4. The approximately Calabi-Yau neck region [051E]

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4. The approximately Calabi-Yau neck region

In this section, we will build the the neck region (or the transition region). It is one of the key geometric ingredients in this paper.

Roughly speaking, we shall construct a family of incomplete Kähler metrics with $S^{1}$ -symmetry, on certain singular $S^{1}$ -fibrations over a cylindrical base. These will serve to interpolate between two different geometries at the ends of two Tian-Yau metrics.

In complex two dimensions, these metrics were constructed in our previous paper [HSVZ18] using the Gibbons-Hawking ansatz applied to the Green’s function on the flat cylinder $\mathbb{T}^{2}\times\mathbb{R}$ . In particular, the resulting metrics are hyperkähler.

In higher dimensions the situation is much more involved. Our construction is motivated by the non-linear Gibbons-Hawking ansatz in Section 2. However, as it was explained in Section 2, it does not seem easy to solve the non-linear reduced equation directly. Instead we shall use a singular solution to the linearized ansatz, namely, the Green’s current constructed in Section 3, to obtain a family of Kähler metrics with $S^{1}$ -symmetry, parametrized by a large parameter $T\gg 1$ . The main differences from the two dimensional case are as follows:

•

These metrics will not be shown to be smooth along the fixed loci of the $S^{1}$ action. Indeed, we will only prove that they are $C^{2,\alpha}$ for all $\alpha\in(0,1)$ . For our gluing construction we shall need a further perturbation which lowers the regularity to be $C^{1,\alpha}$ . This turns out to be sufficient for our analysis.
•

These metrics are not exactly Calabi-Yau. However, we shall show that they are approximately Calabi-Yau, in an appropriate weighted sense (Proposition 4.23). It is possible to perturb these to genuine incomplete Calabi-Yau metrics, see Section 6. But for the proof of our main theorem, in Section 7 we shall directly glue these approximately Calabi-Yau metrics with two pieces of Tian-Yau spaces (c.f. Section 7.2) to form a closed Kähler manifold which is approximately Calabi-Yau, and then apply implicit function theorem.

Let us first set up some notations for this section before moving on. Throughout this section we shall fix integers $n\geq 2$ and $k>0$ .

Let $(D,\omega_{D},\Omega_{D})$ be a compact Calabi-Yau manifold of complex dimension $n-1$ . Here $\omega_{D}$ is a Kähler metric in the class $2\pi c_{1}(L)$ for some ample holomorphic line bundle $L$ , $\Omega_{D}$ is a nowhere vanishing holomorphic volume form on $D$ , and the following normalized Calabi-Yau equation holds,

(4.1)

\frac{1}{(n-1)!}\omega_{D}^{n-1}=\frac{(\sqrt{-1})^{(n-1)^{2}}}{2^{n-1}}\Omega_{D}\wedge\bar{\Omega}_{D}.

We fix a hermitian metric on $L$ whose curvature form is $-\sqrt{-1}\omega_{D}$ . This naturally induces a hermitian metric on any tensor powers of $L$ . We shall also fix a smooth divisor $H$ in the linear system $L^{\otimes k}$ and a defining section $S_{H}$ .

Let

(4.2)

Q\equiv D\times\mathbb{R}

be the Riemannian product, where $\mathbb{R}$ is the real line and parametrized by the coordinate $z\in(-\infty,\infty)$ . We denote

(4.3)

P\equiv H\times\{0\}.

Using the normal exponential map on $D$ (resp. $Q$ ), we may always implicitly identify a tubular neighborhood of $P$ in $D$ (resp. $Q$ ) with a neighborhood of the zero section in the normal bundle $N_{0}$ (resp. $N=N_{0}\oplus\mathbb{R}$ ). Here we adopt the notation in Section 3.3, so $N_{0}$ is a hermitian line bundle and $N$ is the Riemannian vector bundle.

Now we fix $k_{-},k_{+}\in\mathbb{Z}$ with $k_{-}>0$ and $k_{+}<0$ and $k_{-}-k_{+}=k$ . Applying Proposition 3.31, we get a unique Green’s current for $P$ , given in the form

(4.4)

G_{P}=\psi\wedge dz,

such that the asymptotics (3.349) holds.

It turns out that assuming $b_{1}(D)=0$ simplifies the discussion in several places. So we shall always proceed assuming $b_{1}(D)=0$ in this section, and we will make remarks on the general case whenever needed.

To simplify the notations, we also make the following conventions for this section:

•

$\epsilon_{T}$ denotes a family of functions on $D$ , parametrized by $T\geq 1$ , such that for each $k\geq 0$ , its $k$ -th derivative with respect to $\omega_{D}$ is of the form $O(e^{-\delta_{k}T})$ as $T\rightarrow\infty$ , for some $\delta_{k}>0$ (independent of $T$ ).
•

$\underline{\epsilon}_{T}$ denotes a function of $T$ which is $O(e^{-\delta T})$ as $T\rightarrow\infty$ , for some $\delta>0$
•

$\epsilon(z)$ denotes a function on $Q$ such that its all derivatives exponential decay at infinity.
•

$B_{T}$ denotes a family of functions on $D$ , parametrized by $T\gg 1$ , such that for each $k\geq 0$ , its $k$ -th derivatives with respect to $\omega_{D}$ is bounded independent of $T$ .
•

$\underline{B}_{T}$ denotes a function of $T$ which is uniformly bounded as $T\rightarrow\infty$ .
•

$B(z)$ denotes a function of $z$ , such that all its derivatives are uniformly bounded.

The organization of this Section is as follows. In Section 4.1 we use the Green’s currents constructed in Section 3, and the ideas in Section 2 to construct a family of incomplete $S^{1}$ invariant $C^{2,\alpha}$ Kähler structures whose quotient spaces are domains in $Q$ . Special attention are paid to understand the singularity structure near the fixed loci of the $S^{1}$ action. We will first construct a smooth compactification and write an explicit local model, and then study the regularity of the Kähler structures. In Section 4.2 we show the underlying complex manifold is an open subset in an explicit $\mathbb{C}^{*}$ fibration over $D$ , and derive a formula for the Kähler potential of our family of Kähler metrics. In Section 4.3 we study and classify the limit geometry of our family of metrics at regularity scales, which forms a foundation for our weighted analysis. In Section 4.4 we define the relevant weighted Hölder spaces and prove a local weighted Schauder estimate. We also show our family of Kähler metrics are approximately Calabi-Yau by providing an estimate of the error in a weighted Hölder space. In Section 4.5 we deal with a perturbation of the complex structures of the underlying complex manifold, and estimate the error in a weighted Hölder space. This will be used in Section 7. The proof relies on estimating the complex geometric quantities using the weighted Schauder estimates in Section 4.4.

4.1. Construction of a family of $C^{2,\alpha}$ Kähler structures

In this subsection we shall use (2.19) to construct a family of $C^{2,\alpha}$ Kähler structures on certain $S^{1}$ fibrations over increasing domains in $Q$ . So we need to construct a family of pairs $(\tilde{\omega},h)$ parametrized by $T\gg 1$ . Most of the quantities defined in this subsection will depend on the parameter $T$ , but for simplicity of notation we will not always keep track of this if it is clear from the context.

For $T\gg 1$ , we define

(4.5)

\tilde{\omega}=T\omega_{D}+\psi.

It can be viewed as a family of closed $(1,1)$ -forms $\tilde{\omega}(z)$ on $D$ parametrized by $z$ . Using the Kähler identity, we obtain

(4.6)

\partial_{z}^{2}\tilde{\omega}=\Delta_{D}\tilde{\omega}=-d_{D}d_{D}^{c}\Tr_{\omega_{D}}\tilde{\omega},

So if we define

(4.7)

h\equiv\Tr_{\omega_{D}}\tilde{\omega}+q(z)

for any smooth function $q(z)$ , then the pair $(\tilde{\omega},h)$ satisfies the first equation in (2.19):

(4.8)

\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}h=0.

For our purpose we need to make a special choice of the function $q(z)$ . First we define $q_{0}(z)$ by the following co-homological condition

(4.9)

q_{0}(z)\int_{D}\omega_{D}^{n-1}+(n-1)\int_{D}\tilde{\omega}(z)\wedge\omega_{D}^{n-2}=T^{2-n}\int_{D}\tilde{\omega}(z)^{n-1},\ \ \forall z\in\mathbb{R},

By Lemma 3.33, we know that the cohomology class $[\psi(z)]\in H^{2}(D;\mathbb{R})$ is piecewise linear in $z\in\mathbb{R}$ , so

(4.10)

\displaystyle q_{0}(z)=\begin{cases}T^{2-n}(T+k_{+}z)^{n-1}-(n-1)(T+k_{+}z),&z>0,\\ T^{2-n}(T+k_{-}z)^{n-1}-(n-1)(T+k_{-}z),&z<0.\end{cases}

It follows that $q_{0}(z)$ is identically zero if $n=2$ , which corresponds to the case of the classical Gibbons-Hawking anstaz used in [HSVZ18]. But if $n>2$ then $q_{0}(z)$ is only $C^{1,1}$ at $z=0$ and we need to smooth it. We shall fix throughout this section a smooth function $L_{0}:\mathbb{R}\rightarrow\mathbb{R}$ satisfying

(4.11)

\displaystyle L_{0}(z)\equiv\begin{cases}k_{+}z,&z>1,\\ 0,&z=0,\\ k_{-}z,&z<-1.\end{cases}

and let

(4.12)

L_{T}(z)\equiv T+L_{0}(z).

Then we define

(4.13)

q(z)\equiv T^{2-n}L_{T}(z)^{n-1}-(n-1)L_{T}(z).

It follows that $q(z)$ is smooth and agrees with $q_{0}(z)$ when $|z|\geq 1$ . It is also easy to see that correspondingly we have

(4.14)

\displaystyle\int_{D}h\omega_{D}^{n-1}=\begin{cases}T^{2-n}\int_{D}\tilde{\omega}(z)^{n-1},&|z|\geq 1,\\ T^{2-n}\int_{D}\tilde{\omega}(z)^{n-1}+T^{-1}B(z),&z\in[-1,1].\end{cases}

We refer to Remark 4.3.1 for an explanation of this choice of $q(z)$ .

To apply the construction in Section 2, we need to restrict to the region in $Q$ where $\tilde{\omega}(z)$ is a positive form and $h$ is a positive function. For $T$ large we define $T_{+}>0$ and $T_{-}<0$ by

(4.15)

\begin{cases}T+k_{+}T_{+}=T^{\frac{n-2}{n}}\\ T+k_{-}T_{-}=T^{\frac{n-2}{n}}.\end{cases}

and denote by $Q_{T}\subset Q$ the region where $z\in[T_{-},T_{+}]$ .

Lemma 4.1.

For $T$ large, over $Q_{T}\setminus P$ , both $\tilde{\omega}$ and $h$ are positive. Moreover, $h$ has the following approximation formula

(4.16)		$\displaystyle h$	$\displaystyle=T^{2-n}(T+k_{\pm}z)^{n-1}+\epsilon(z),\quad\|z\|\geq 1,$
(4.17)		$\displaystyle h$	$\displaystyle=T+\frac{1}{2r}+O^{\prime}(r)+T^{-1}B(z),\quad\|z\|\leq 1,$

where $O^{\prime}(r)$ is a fixed function independent of $T$ , and it has the singular behavior near $P$ given by Definition 3.3.

Proof.

We first consider $\tilde{\omega}$ . As $z\rightarrow\pm\infty$ the behavior of $\tilde{\omega}$ is governed by (3.349), so for $T\gg 1$ we know $\tilde{\omega}$ is positive over the region where $z\in[-k_{-}^{-1}(T-1),-k_{+}^{-1}(T-1)]\setminus[-C,C]$ for some number $C>0$ independent of $T$ . By the expansion of $\psi$ in a neighborhood of $P$ given in Proposition 3.24, for $T$ sufficiently large, $\tilde{\omega}$ is also positive when $z\in[-C,C]$ . Hence $\tilde{\omega}$ is positive over the region where $z\in[-k_{-}^{-1}(T-1),-k_{+}^{-1}(T-1)].$ Since this contains $Q_{T}$ we see in particular $\tilde{\omega}$ is positive over $Q_{T}\setminus P$ .

To deal with $h$ we need to analyze $q(z)$ . When $|z|\geq 1$ , we have

(4.18)

q(z)=q_{0}(z)=T^{2-n}(T+k_{\pm}z)^{n-1}-(n-1)(T+k_{\pm}z),

where the choice of $+$ or $-$ depends on whether $z>0$ or $z<0$ . By (3.349) we then get

(4.19)

h=T^{2-n}(T+k_{\pm}z)^{n-1}+\epsilon(z).

So we can find $C>0$ such that $h$ is positive when $z\in[T_{-},T_{+}]\setminus[-C,C]$ . On the other hand, on $[-C,C]$ we know by definition

(4.20)

q(z)=(2-n)T+T^{-1}B(z).

Hence by the expansion in Proposition 3.28 we obtain (4.17). This implies that for $T\gg 1$ , $h$ is also positive when $z\in[-C,C]$ . ∎

Now we define the 2-form

(4.21)

\Upsilon\equiv\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h.

Then (4.6) implies that $\Upsilon$ is closed on $Q\setminus P$ and hence $[\Upsilon]\in H^{2}(Q\setminus P,\mathbb{R})$ . Moreover, we have

Lemma 4.2.

The cohomology class $\frac{1}{2\pi}[\Upsilon]\in H^{2}(Q\setminus P;\mathbb{R})$ is integral.

Proof.

As mentioned in the beginning of this section, we identify a tubular neighborhood of $P$ in $Q$ with a neighborhood of the zero section in its normal bundle $N=N_{0}\oplus\mathbb{R}$ . For simplicity we may assume this neighborhood is given by $\mathcal{B}_{\epsilon}$ , the 2-ball bundle over $P$ consisting of the set of all elements in $N_{0}\oplus\mathbb{R}$ with norm smaller than or equal to $\epsilon$ , and we denote by $\mathcal{S}_{\epsilon}$ the boundary of $\mathcal{B}_{\epsilon}$ .

Fix $z_{0}>0$ , then the composition of the natural maps

(4.22)

D\simeq D\times\{z_{0}\}\hookrightarrow Q\setminus P\hookrightarrow Q\rightarrow D

is the identity map, which implies that for all $k$ , the map $H_{k}(Q\setminus P;\mathbb{Z})\rightarrow H_{k}(Q;\mathbb{Z})$ is surjective and we have a natural splitting

(4.23)

H_{2}(Q\setminus P;\mathbb{Z})=H_{2}(D;\mathbb{Z})\oplus K

for some $K$ . By assumption for $z>0$ ,

(4.24)

[\partial_{z}\tilde{\omega}(z)]=[\partial_{z}\psi(z)]=k_{+}[\omega_{D}]=2\pi k_{+}c_{1}(L),

so $\frac{1}{2\pi}[\Upsilon]|_{D\times\{z_{0}\}}=k_{+}c_{1}(L)$ is integral. Hence it suffices to show the integral of $\frac{1}{2\pi}\Upsilon$ over any element in $K$ is also an integer.

By the Mayer-Vietoris sequence applied to $Q=(Q\setminus P)\cup\mathcal{B}_{\epsilon}$ , we get

(4.25)

0\rightarrow H_{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\rightarrow H_{2}(Q\setminus P;\mathbb{Z})\oplus H_{2}(\mathcal{B}_{\epsilon};\mathbb{Z})\rightarrow H_{2}(Q;\mathbb{Z})\simeq H_{2}(D;\mathbb{Z})\rightarrow 0.

So we obtain the exact sequence

(4.26)

0\rightarrow K\rightarrow H_{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\rightarrow H_{2}(\mathcal{B}_{\epsilon};\mathbb{Z})\simeq H_{2}(P;\mathbb{Z}).

On the other hand, by the Gysin sequence applied to the 2-sphere bundle $p:\mathcal{S}_{\epsilon}\rightarrow P$ we get

(4.27)

0\rightarrow H^{2}(P;\mathbb{Z})\xrightarrow{p^{*}}H^{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\xrightarrow{\int}H^{0}(P;\mathbb{Z})\xrightarrow{\wedge e}H^{3}(P;\mathbb{Z})\rightarrow\cdots

where $\int$ denotes integration over the 2-sphere fibers, and $\wedge e$ denotes the wedge product with Euler class of $\mathcal{S}_{\epsilon}$ . Since the Euler class $e$ of $N_{0}\oplus\mathbb{R}$ vanishes, the above becomes

(4.28)

0\rightarrow H^{2}(P;\mathbb{Z})\xrightarrow{p^{*}}H^{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\xrightarrow{\int}H^{0}(P;\mathbb{Z})\simeq\mathbb{Z}\rightarrow 0.

(4.26) and (4.28) together imply that modulo torsion, $K$ is generated by the homology class of a 2-sphere fiber of $p$ . So we just need to show $\int\frac{1}{2\pi}[\Upsilon]|_{\mathcal{S}_{\epsilon}}$ is an integer.

By the expansion of $\psi$ and $h$ in Proposition 3.24 and Proposition 3.28, it is easy to check that by restricting to the fiber of $N$ over $p$ , we have

(4.29)

\Upsilon|_{N(p)}=-\frac{\sqrt{-1}}{4r^{3}}(zdyd\bar{y}+(yd\bar{y}-\bar{y}dy)dz)+O(1).

Further restricting to the $2$ -sphere with radius $\epsilon$ , we get

(4.30)

\Upsilon|_{\mathcal{S}_{\epsilon}(p)}=-\frac{1}{2\epsilon^{2}}\dvol_{S^{2}_{\epsilon}}+O(1),

where $\dvol_{S^{2}_{\epsilon}}$ is the area form of the standard $\epsilon$ -sphere in $\mathbb{R}^{3}$ . Taking the integral and let $\epsilon\rightarrow 0$ gives that

(4.31)

\int_{\mathcal{S}_{\epsilon}(p)}\Upsilon=-2\pi.

∎

By Lemma 4.2, standard theory yields a $U(1)$ connection $1$ -form $-\sqrt{-1}\Theta$ on a principal $S^{1}$ -bundle

(4.32)

\pi:\mathcal{M}^{*}\rightarrow Q_{T}\setminus P

with curvature form $-\sqrt{-1}\Upsilon$ . Moreover, $\mathcal{M}^{*}$ restricts to the standard Hopf bundle on each normal $S^{2}$ to $P$ (it has degree $-1$ if we use the natural orientation). Then we have the second equation in (2.19) satisfied:

(4.33)

d\Theta=\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h.

On ${\mathcal{M}^{*}}$ we define a real-valued 2-form

(4.34)

\omega\equiv T^{\frac{2-n}{n}}(\pi^{*}\tilde{\omega}+dz\wedge\Theta)

and a complex-valued $n$ -form

(4.35)

\Omega\equiv\sqrt{-1}(hdz+\sqrt{-1}\Theta)\wedge\pi^{*}\Omega_{D}.

One can directly check that both $\omega$ and $\Omega$ are closed. By the discussion in Section 2, we know $(\omega,\Omega)$ defines a smooth Kähler metric on ${\mathcal{M}^{*}}$ , so that $\Omega$ is the holomorphic volume form and $\omega$ is the Kähler form. Also $h^{-1}$ has an intrinsic geometric meaning as the norm squared of the Killing field generating the $S^{1}$ action.

By (4.1) and straightforward calculations, we have

(4.36)

\frac{(\sqrt{-1})^{n^{2}}2^{-n}\Omega\wedge\bar{\Omega}}{\omega^{n}/n!}=\frac{T^{-1}h\omega_{D}^{n-1}}{(\omega_{D}+T^{-1}\psi)^{n-1}}.

Definition 4.3.

Given the above constructed Kähler metric $\omega$ , the error function is defined by

(4.37)

\mathrm{Err}_{CY}\equiv\frac{T^{-1}h\omega_{D}^{n-1}}{(\omega_{D}+T^{-1}\psi)^{n-1}}-1.

In particular, $\omega$ is a Calabi-Yau metric if $\mathrm{Err}_{CY}=0$ .

Remark 4.3.1.

Now we are ready to explain the reason for the choice of the function $q(z)$ and the rescaling factor $T^{\frac{n-2}{n}}$ in the above definition of $\omega$ . These are chosen to make the Kähler metric $(\omega,\Omega)$ approximately Calabi-Yau in the following sense:

(1)

Applying (3.349) and (4.16), we have $\mathrm{Err}_{CY}=T^{-2}\epsilon(z(\bm{x}))$ for $\bm{x}\in\mathcal{M}^{*}$ satisfying $|z(\bm{x})|\geq C$ .
(2)

Applying (3.316) and (4.17), we have $\mathrm{Err}_{CY}=O(T^{-2})$ for $\bm{x}\in\mathcal{M}^{*}$ satisfying $|z(\bm{x})|\leq C$ and $d_{Q}(\bm{x},P)\geq d_{0}>0$ , where $d_{0}>0$ is some definite constant.

We will need a more precise weighted estimate on $\mathrm{Err}_{CY}$ . See Proposition 4.23.

Remark 4.3.2.

As explained in Section 2, a priori these structures depend on the choice of $\Theta$ . But we claim that in our current setting $b_{1}(D)=0$ , the choice of $\Theta$ will not change the isomorphism class of the Kähler structures. Given two choices $\Theta$ and $\Theta^{\prime}$ , then the difference $\Theta^{\prime}-\Theta$ is a closed 1-form on $Q_{T}\setminus P$ . Since $P$ has codimension $3$ in $Q$ , we know $H^{1}(Q_{T}\setminus P;\mathbb{R})\simeq H^{1}(Q;\mathbb{R})\simeq H^{1}(D;\mathbb{R})$ . Hence we can write

(4.38)

\Theta^{\prime}-\Theta=df+\beta

for a function $f$ on $Q_{T}\setminus P$ and a harmonic 1-form $\beta$ on $D$ . So if $b_{1}(D)=0$ then $\beta=0$ , and the isomorphism class of the Kähler structure $(\omega,\Omega)$ does not depend on the choice of $\Theta$ . In the general case when $b_{1}(D)>0$ , up to gauge equivalence, $\Theta$ and $\Theta^{\prime}$ differ by the pull-back of a flat connection on $D$ . In Remark 4.8.2 we shall see the geometric meaning of this.

Next we move on to the study the compactified geometry of ${\mathcal{M}^{*}}$ near $P$ . We shall first construct a smooth model for the compactification and then study the regularity of the Kähler metric on this model.

As before we will always identify a neighborhood $\mathcal{U}$ of $P$ in $Q$ with a tubular neighborhood of the zero section in $N_{0}\oplus\mathbb{R}$ over $H$ . Denote by $\mathbb{L}_{1}$ and $\mathbb{L}_{2}$ the complex line bundles over $H$ given by the restriction

(4.39)

\mathbb{L}_{1}\equiv L^{\otimes-k_{+}}|_{H};\ \ \mathbb{L}_{2}\equiv L^{\otimes k_{-}}|_{H}.

Then as complex line bundles $N_{0}$ is isomorphic to $L^{\otimes k}|_{H}\simeq\mathbb{L}_{1}\otimes\mathbb{L}_{2}$ , and we fix such an isomorphism now. Notice $N_{0}$ is equipped with a natural hermitian metric induced from the Kähler metric $\omega_{D}$ on $D$ (c.f. Section 3.3). This then determines a hermitian metric on $L$ hence on $\mathbb{L}_{1}$ and $\mathbb{L}_{2}$ . Define

(4.40)

\mathbb{L}\equiv\mathbb{L}_{1}\oplus\mathbb{L}_{2},

and consider the map

(4.41)

\tau:\mathbb{L}\rightarrow N_{0}\oplus\mathbb{R};(s_{1},s_{2})\mapsto(s_{1}\otimes s_{2},\frac{|s_{1}|^{2}-|s_{2}|^{2}}{2}).

Away from the zero section in $\mathbb{L}$ , $\tau$ is a principal $S^{1}$ bundle, with the $S^{1}$ action given by

(4.42)

e^{\sqrt{-1}\mathfrak{t}}\cdot(s_{1},s_{2})=(e^{-\sqrt{-1}t}s_{1},e^{\sqrt{-1}t}s_{2}).

As Section 3.3, locally choosing holomorphic coordinates $\{w_{1},\cdots,w_{n-1}\}$ on $D$ centered at $p\in H$ . These give rise to local coordinates $\{y,\bar{y},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ on $N_{0}$ , and also a local unitary section of $N_{0}$ in the form $\bm{e}=|\sigma|^{-1}\cdot\sigma$ . Then we choose a local section $\bm{e}_{L}$ of $L|_{H}$ with $\bm{e}_{L}^{\otimes k}=\bm{e}$ . Correspondingly we get local unitary sections $\bm{e}_{1}\equiv\bm{e}_{L}^{\otimes-k_{+}},\bm{e}_{2}\equiv\bm{e}_{L}^{\otimes k_{-}}$ of $\mathbb{L}_{1},\mathbb{L}_{2}$ respectively. Then we obtain local fiber coordinates $u_{1},u_{2},y$ on $\mathbb{L}_{2},\mathbb{L}_{1},N_{0}$ respectively by writing

(4.43)

s_{1}=u_{1}\bm{e}_{1},s_{2}=u_{2}\bm{e}_{2},s=y\bm{e}.

Then the map $\tau$ can be represented in coordinates as

(4.44)

\begin{cases}y=u_{1}u_{2}\\ z=\frac{1}{2}(|u_{1}|^{2}-|u_{2}|^{2})\end{cases}

Hence $\tau$ is the standard Hopf fibration $\mathbb{C}^{2}\rightarrow\mathbb{R}^{3}$ over each fiber.

Lemma 4.4.

Over $\mathcal{U}\setminus P$ , the principal $S^{1}$ bundle ${\mathcal{M}^{*}}$ is isomorphic to $\mathbb{L}$ .

Proof.

Notice a principal $S^{1}$ bundle is topologically determined by its first Chern class. It suffices to compare the first Chern classes of ${\mathcal{M}^{*}}$ and $\mathbb{L}$ over the sphere bundle $\mathcal{S}_{\epsilon}$ for a small $\epsilon$ . As in the proof of Lemma 4.2 th Gysin sequence gives

(4.45)

0\rightarrow H^{2}(P;\mathbb{Z})\xrightarrow{p^{*}}H^{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\xrightarrow{\int}H^{0}(P;\mathbb{Z})\simeq\mathbb{Z}\rightarrow 0.

From the proof of Lemma 4.2 we know

(4.46)

\int c_{1}({\mathcal{M}^{*}})=\int_{\mathcal{S}_{\epsilon}(p)}\frac{1}{2\pi}\Upsilon=-1.

Also by (2.49) we have

(4.47)

\int c_{1}(\mathbb{L})=\int_{S^{2}\subset\mathbb{R}^{3}}\frac{1}{2\pi}\Upsilon_{0}=-1.

(4.48)

{\mathcal{M}^{*}}=\mathbb{L}\otimes p^{*}L^{\prime}

for some $U(1)$ bundle $L^{\prime}$ over $P$ . Now we restrict both ${\mathcal{M}^{*}}$ and $\mathbb{L}$ to the subset $H_{0}\subset\mathcal{U}$ where $y=0$ and $z=z_{0}$ for a fixed $z_{0}<0$ . We can identify $H_{0}$ with $H$ by the projection map. Now we claim both restrictions have first Chern class equal to $k_{-}c_{1}(\mathbb{L}_{2})$ . For ${\mathcal{M}^{*}}$ this follows from construction and for $\mathbb{L}$ we notice that $z=z_{0}<0$ implies that $s_{2}\neq 0$ and $s_{1}=0$ , so the projection map $(s_{1},s_{2})\mapsto|2z_{0}|^{1/2}\cdot s_{2}$ gives an isomorphism between the restriction of $\mathbb{L}$ and the unit circle bundle in $\mathbb{L}_{2}$ . This also explains the choice of the weight of the $S^{1}$ action in (4.42).

Now it follows from the claim that $L^{\prime}$ is indeed a trivial principal $S^{1}$ bundle, and this finishes the proof. ∎

By Lemma 4.4 we may glue ${\mathcal{M}^{*}}$ and $\mathbb{L}$ together to obtain a differentiable compactfication $\mathcal{M}$ of ${\mathcal{M}^{*}}$ . The projection map $\pi$ naturally extends to a map

(4.49)

\pi:\mathcal{M}\rightarrow Q_{T}

which is a singular $S^{1}$ fibration, with discriminant locus given by $P$ . We shall identify

(4.50)

\mathcal{P}\equiv\pi^{-1}(P)

with the zero section in $\mathbb{L}$ , and identify a neighborhood of $\mathcal{P}$ with a neighborhood of the zero section in $\mathbb{L}$ and the projection map $\pi$ with the above $\tau$ .

To study the regularity of the Kähler metric $(\omega,\Omega)$ on the compactification $\mathcal{M}$ , we shall make a special choice of the connection 1-form $-\sqrt{-1}\Theta$ on a neighborhood $\mathcal{V}$ of $\mathcal{P}$ in $\mathbb{L}$ , with curvature form $\Upsilon$ , which has explicit regularity behavior across $\mathcal{P}$ . To do this, we need a few steps. First, we notice that $\{u_{1},\bar{u}_{1},u_{2},\bar{u}_{2},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ provides local coordinates on $\mathbb{L}$ , and we can define a local model connection 1-form on $\mathbb{L}$ by simply taking the model formula (2.47):

(4.51)

\Theta_{0}=-\sqrt{-1}\frac{\bar{u}_{1}du_{1}-u_{1}d\bar{u}_{1}-\bar{u}_{2}du_{2}+u_{2}d\bar{u}_{2}}{2(|u_{1}|^{2}+|u_{2}|^{2})}.

Just as in the discussion in Section 2, we see $\Theta_{0}(\partial_{t})=-1$ , where $\partial_{t}$ is the vector field generating the $S^{1}$ action. It is clear that the definition of $\Theta_{0}$ only depends on the choice of $\sigma$ and does not depend on the choice of $\bm{e}_{1}$ and $\bm{e}_{2}$ (which has the freedom of multiplying by a constant root of unity).

To make a globally defined connection 1-form, we need to add a correction term, and define

(4.52)

\Theta_{1}=\Theta_{0}+\frac{z}{r}\Gamma-\frac{k_{-}+k_{+}}{k_{-}-k_{+}}\Gamma,

where $\Gamma$ is the local 1-form given in Section 3.3, and we have implicitly viewed forms on $H$ as forms on $\mathbb{L}$ using the pull-back $\pi^{*}$ .

Proposition 4.5.

$-\sqrt{-1}\Theta_{1}$ is a globally-defined connection 1-form on the $S^{1}$ bundle $\tau:\mathbb{L}\setminus\mathcal{P}\rightarrow N\setminus H$ , and we have

(4.53)

d\Theta_{1}-\Upsilon=O^{\prime}(s),

where

(4.54)

s^{2}\equiv|u_{1}|^{2}+|u_{2}|^{2}=2r,

and we have adopted the $O^{\prime}$ notation in Section 3.1 for the submanifold $\mathcal{P}\subset\mathbb{L}$ .

Proof.

To see $\Theta_{1}$ is a well-defined, we consider the change of unitary frame $\bm{e}$ on $N_{0}$ to $\tilde{\bm{e}}=e^{\sqrt{-1}k\phi}\bm{e}$ , then we have

(4.55)

\tilde{y}=e^{-(k_{-}-k_{+})\sqrt{-1}\phi}y;\ \ \tilde{u}_{1}=e^{k_{+}\sqrt{-1}\phi}u_{1},\tilde{u}_{2}=e^{-k_{-}\sqrt{-1}\phi}u_{2}.

for some local real-valued function $\phi$ on $H$ . Then we get

(4.56)	$\displaystyle\bar{u}_{1}du_{1}-u_{1}d\bar{u}_{1}$	$\displaystyle=\bar{\tilde{u}}_{1}d\tilde{u}_{1}-\tilde{u}_{1}d\bar{\tilde{u}}_{1}-2k_{+}\sqrt{-1}\|u_{1}\|^{2}d\phi,$
(4.57)	$\displaystyle\bar{u}_{2}du_{2}-u_{2}d\bar{u}_{2}$	$\displaystyle=\bar{\tilde{u}}_{2}d\tilde{u}_{2}-\tilde{u}_{2}d\bar{\tilde{u}}_{2}+2k_{-}\sqrt{-1}\|u_{2}\|^{2}d\phi,$
(4.58)	$\displaystyle\Gamma$	$\displaystyle=\widetilde{\Gamma}-\frac{k_{-}-k_{+}}{2}d\phi.$

Then it is a straightforward to compute that $\widetilde{\Theta}_{1}=\Theta_{1}$ , which shows that $\Theta_{1}$ is globally defined.

Now we consider the local expansion of $\Upsilon$ . First differentiating the expansion of $\psi$ in Proposition 3.24 we get

(4.59)

\partial_{z}\tilde{\omega}=-\sqrt{-1}\frac{z}{2r^{3}}dy\wedge d\bar{y}-\frac{z}{2r^{3}}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+\frac{z}{r}d\Gamma+O^{\prime}(1)dy+O^{\prime}(1)d\bar{y}+O^{\prime}(r).

Next, applying Proposition 3.28 and Proposition 3.26, we obtain

(4.60)

d_{D}^{c}h=d_{D}^{c}(\frac{1}{2r}+O^{\prime}(r))=-\frac{1}{4r^{3}}d_{D}^{c}|y|^{2}+O^{\prime}(1)=-\frac{\sqrt{-1}(yd\bar{y}-\bar{y}dy)+4|y|^{2}\Gamma}{4r^{3}}+O^{\prime}(1).

Putting together these, and noting that $d\Theta_{0}$ is given as in (2.50), we obtain

(4.61)

d\Theta_{1}-\Upsilon=O^{\prime}(1)dy+O^{\prime}(1)d\bar{y}+O^{\prime}(r)+O^{\prime}(1)dz.

Now translating into the coordinates $u_{1},u_{2}$ on $\mathbb{L}$ we obtain the conclusion.

∎

Remark 4.5.1.

It follows that $d\Theta_{1}$ and $\Upsilon$ are cohomologous on a tubular neighborhood of $\mathcal{P}$ in $\mathbb{L}$ . One can also see this by a direct calculation. For example, by restricting to a slice with $z>0$ and $y=0$ , it is clear by Lemma 3.33 we know $\Upsilon$ is cohomologous to $k_{+}\omega_{D}|_{H}$ . On the other hand, by definition $d\Theta_{1}$ on this slice is given by $(1-\frac{k_{+}+k_{-}}{k_{-}-k_{+}})d\Gamma=k_{+}\omega_{D}$ (using Lemma 3.25)

The next Lemma allows us to correct $O^{\prime}(s)$ term on the right hand side. We fix any $S^{1}$ invariant Riemannian metric on $\mathbb{L}$ .

Lemma 4.6.

There exists a local 1-form $\theta$ on a neighborhood of $\mathcal{P}$ in $\mathbb{L}$ with the following properties:

(1)

$\theta=O^{\prime}(s^{2})$ ,
(2)

$\theta$ is smooth away from $\pi^{-1}(P)$ ,
(3)

$\mathcal{L}_{\partial_{t}}\theta=0$ ,
(4)

$\partial_{t}\lrcorner\theta=0$ ,
(5)

$d(\Theta_{1}+\theta)=\Upsilon$ .

Proof.

From the above Remark we know $d\Theta_{1}-\Upsilon$ is cohomologous to zero. The existence of a solution $\theta$ to $d(\Theta_{1}+\theta)=\Upsilon$ is obtained by adding the gauge fixing condition $d^{*}\theta=0$ , and solving the elliptic system with Neumann boundary condition

(4.62)

\begin{cases}d\theta=\Upsilon-d\Theta_{1},\\ d^{*}\theta=0,\\ \theta(\nu)=0,\ \ \text{on}\ \ \partial\mathcal{V}.\end{cases}

on a tubular neighborhood $\mathcal{V}$ of $\mathcal{P}$ in $\mathbb{L}$ . See Proposition 3.7 in [DS14] for example. By Proposition 4.5 we know $\Upsilon-d\Theta_{1}=O^{\prime}(s)$ , particularly, $\Upsilon-d\Theta_{1}\in C^{\alpha}$ for all $\alpha\in(0,1)$ . Hence standard elliptic regularity guarantees a solution $\theta\in C^{1,\alpha}$ and is smooth away from $\mathcal{P}$ . Since both $\Upsilon$ and $\Theta_{1}$ are $S^{1}$ -invariant, by averaging we may assume $\theta$ is $S^{1}$ -invariant too, hence $\mathcal{L}_{\partial_{t}}\theta=0$ on the smooth part. Also since $\Upsilon$ and $d\Theta_{1}$ are pulled-back from the base $Q_{T}\setminus P$ , we have

(4.63)

\partial_{t}\lrcorner\Upsilon=\partial_{t}\lrcorner d\Theta_{1}=0.

So we get

(4.64)

d(\partial_{t}\lrcorner\theta)=\mathcal{L}_{\partial_{t}}\theta-\partial_{t}\lrcorner(d\theta)=0.

This implies $\partial_{t}\lrcorner\theta$ is a constant. Now as we approach $\mathcal{P}$ , the norm of $\partial_{t}$ , with respect to the fixed metric on $\mathbb{L}$ , must go to zero, hence we see

(4.65)

\partial_{t}\lrcorner\theta=0.

The higher regularity of $\theta$ follows just as in the proof of Lemma 3.22 in Section 3. ∎

Now we define a fixed connection 1-form on $\mathbb{L}$ .

(4.66)

\Theta_{m}\equiv\Theta_{1}+\theta,

Therefore, in a neighborhood of $\mathcal{P}\subset\mathbb{L}$ minus $\mathcal{P}$ , the original choice of $\Theta$ can be written as

(4.67)

\Theta=\Theta_{m}+\theta_{f},

where $\theta_{f}$ is a flat connection, which is gauge equivalent to the pull-back of a flat connection on $D$ . Without loss of generality, we can then assume $\theta_{f}$ is smooth.

Proposition 4.7.

With respect to the choice of the connection form $\Theta$ given in (4.67), $(\omega,\Omega)$ defined by (4.34) and (4.35) gives a $C^{2,\alpha}$ (for all $\alpha\in(0,1)$ ) Kähler structure on $\mathbb{L}$ which is invariant under the natural $S^{1}$ -action and is smooth outside $\mathcal{P}$ .

Proof.

At the first stage, we will analyze the regularity of $\omega$ . By definition,

(4.68)

T^{\frac{n-2}{n}}\omega=T\pi^{*}\omega_{D}+\pi^{*}\psi+dz\wedge\Theta.

To start with, let us compute the lifting $\pi^{*}\psi$ . By (3.264),

(4.69)

\pi^{*}\psi=\pi^{*}\tilde{\omega}_{0}+\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+rd\Gamma+\pi^{*}(O^{\prime}(r)dy+O^{\prime}(r)d\bar{y})+\pi^{*}O^{\prime}(r^{2}),

where

(4.70)

\tilde{\omega}_{0}=\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y}

is the standard form in the model setting (2.45). We also notice that

(4.71)		$\displaystyle\pi^{*}(O^{\prime}(r)dy+O^{\prime}(r)d\bar{y})$	$\displaystyle=sO^{\prime}(s^{2}),$
(4.72)		$\displaystyle\pi^{*}O^{\prime}(r^{2})$	$\displaystyle=O^{\prime}(s^{4}).$

Now by definition

(4.73)

\Theta=\Theta_{0}+\frac{z}{r}\Gamma+\frac{k_{-}+k_{+}}{k_{-}-k_{+}}\Gamma+\theta+\theta_{f}=\Theta_{0}+\frac{z}{r}\Gamma+O^{\prime}(s^{2}).

Moreover, according to the discussions in Section 2, we have

(4.74)

\pi^{*}\tilde{\omega}_{0}+dz\wedge\Theta_{0}=\omega_{\mathbb{C}^{2}},

where $\omega_{\mathbb{C}^{2}}=\frac{\sqrt{-1}}{2}(du_{1}\wedge d\bar{u}_{1}+du_{2}\wedge d\bar{u}_{2})$ is the standard Kähler form of $\mathbb{C}^{2}$ . Therefore,

(4.75)

\displaystyle\pi^{*}\psi+dz\wedge\Theta=

\displaystyle\omega_{\mathbb{C}^{2}}+rd\Gamma+\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+dz\wedge(\frac{z}{r}\Gamma)+O^{\prime}(s^{3}).

Using the relation $r^{2}=|y|^{2}+z^{2}$ and the simple computation

(4.76)

d(r\Gamma)=rd\Gamma+dr\wedge\Gamma=rd\Gamma+\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+dz\wedge(\frac{z}{r}\Gamma),

we have

	$\displaystyle\pi^{*}\psi+dz\wedge\Theta=$	$\displaystyle\omega_{\mathbb{C}^{2}}+d(r\Gamma)+O^{\prime}(s^{3})$
(4.77)		$\displaystyle=$	$\displaystyle\omega_{\mathbb{C}^{2}}+O^{\prime}(s^{3}),$

where we use the fact that $r=\frac{1}{2}s^{2}$ and hence $r\Gamma=s^{2}\Gamma$ is smooth on $\mathbb{L}$ . Then it follows that

(4.78)

T^{\frac{n-2}{n}}\omega=T\pi^{*}\omega_{D}+\omega_{\mathbb{C}^{2}}+O^{\prime}(s^{3}).

Hence we see the $(1,1)$ -form $\omega$ locally extends to a $C^{2,\alpha}$ -form across the subset $\{u_{1}=u_{2}=0\}$ .

Now we analyze the regularity of the holomorphic volume form $\Omega$ which is given by

(4.79)

\Omega=\sqrt{-1}(hdz+\sqrt{-1}\Theta)\wedge\pi^{*}\Omega_{D}.

By Lemma 3.30, locally we have

(4.80)

\pi^{*}\Omega_{D}=F(u_{1}du_{2}+u_{2}du_{1}+2\sqrt{-1}u_{1}u_{2}\Gamma)\wedge\pi^{*}\Omega_{H}+\widetilde{O}(s^{2})(u_{1}du_{2}+u_{2}du_{1})+\widetilde{O}(s^{3}).

Also

(4.81)

hdz+\sqrt{-1}\Theta=q(z)dz+\frac{1}{|u_{1}|^{2}+|u_{2}|^{2}}(-\bar{u}_{2}du_{2}+\bar{u}_{1}du_{1}+\sqrt{-1}(|u_{1}|^{2}-|u_{2}|^{2})\Gamma)+O^{\prime}(s^{2}).

Therefore,

(4.82)

\Omega=Fdu_{1}\wedge du_{2}\wedge\Omega_{H}+\sqrt{-1}F(u_{2}du_{1}-u_{1}du_{2})\wedge\Gamma\wedge\Omega_{H}+\widetilde{O}(s^{2})+sO^{\prime}(s^{2}).

This implies that $\Omega$ also extends to a $C^{2,\alpha}$ form across $\{u_{1}=u_{2}=0\}$ . This is equivalent to saying that the almost complex structure $J$ determined by $\Omega$ extends to a $C^{2,\alpha}$ almost complex structure on $\mathcal{M}$ . ∎

Using the Newlander-Nirenberg theorem , we may find locally $C^{3,\alpha}$ holomorphic coordinates, making the complex structure locally standard while still keeping the Kähler form in the class $C^{2,\alpha}$ .

By construction the Kähler structure $(\omega,\Omega)$ is preserved by the natural $S^{1}$ action. The corresponding Killing field is given by

(4.83)

\partial_{t}=-\sqrt{-1}(u_{1}\partial_{u_{1}}-u_{2}\partial_{u_{2}})+\sqrt{-1}(\bar{u}_{1}\partial_{\bar{u}_{1}}-\bar{u}_{2}\partial_{\bar{u}_{2}}).

The zero set $\mathcal{P}$ is a complex submanifold of $\mathcal{M}$ which bi-holomorphic to $H\subset D$ . We also dnote the corresponding holomorphic vector field

(4.84)

\xi^{1,0}=\frac{1}{2}(\partial_{t}-\sqrt{-1}J\partial_{t}).

We also have a smooth holomorphic projection $\pi:\mathcal{M}\rightarrow D\setminus H$ whose fibers are holomorphic cylinders (isomorphic to annuli in $\mathbb{C}$ ). In the next subsection we shall understand the underlying complex manifold and the Kähler potentials on $\mathcal{M}$ .

4.2. Kähler geometry

A key feature in the analysis in Kähler geometry is that we can describe the geometry in terms of a single potential function. This has led to a vast simplification of formulae in Kähler geometry as compared to more general Riemannian geometric setting, and it also has allowed various techniques from PDE and several complex variables, etc to be exploited.

The goal of this subsection is to derive a formulae for the Kähler potential for our Kähler manifold $(\mathcal{M},\omega,\Omega)$ . This is one of the most crucial observations in this paper.

In Section 4.2.1 we will identify the underlying complex manifold of the family of Kähler metrics constructed in the Section 4.1 as a family of open subsets of a fixed complex manifold. In Section 4.2.2 we derive a formula for the Kähler potential.

4.2.1. The underlying complex manifold

We define the following holomorphic line bundles on $D$

(4.85)

L_{+}\equiv L^{-\otimes k_{+}},\ L_{-}\equiv L^{\otimes k_{-}}.

Denote by $\mathcal{N}^{0}$ the hypersurface in the total space of $L_{+}\oplus L_{-}$ defined by the equation

(4.86)

\zeta_{+}\otimes\zeta_{-}=S_{H}(x),

where $\zeta_{\pm}$ denotes points on the fibers of $L_{\pm}$ over $x\in D$ . Since $H$ is smooth, $\mathcal{N}^{0}$ is also smooth, and the submanifold

(4.87)

\mathcal{H}\equiv\{\zeta_{+}=\zeta_{-}=0\}

is naturally isomorphic to $H$ . The fixed hermitian metric on $L$ then induces hermitian metrics on $L_{\pm}$ , which yields the norm functions on $L_{\pm}$ :

(4.88)

r_{\pm}(\zeta_{\pm})\equiv\|\zeta_{\pm}\|.

Then by the projection of $\mathcal{N}^{0}$ to $L_{\pm}$ we may also view $r_{\pm}$ as functions on $\mathcal{N}^{0}$ .

There is a natural holomorphic volume form on $\mathcal{N}^{0}$ given by

(4.89)

\Omega_{\mathcal{N}^{0}}\equiv\frac{\sqrt{-1}}{2}(\frac{d\zeta_{+}}{\zeta_{+}}-\frac{d\zeta_{-}}{\zeta_{-}})\wedge\Omega_{D}

where $\Omega_{D}$ means the pull-back of $\Omega_{D}$ to $\mathcal{N}^{0}$ and for simplicity of notation we shall omit the pull-back notation when the meaning is clear from the context. The expression on the right hand side of (4.89) should be understood in the following sense: after choosing a local holomorphic frame $\sigma$ of $L$ , $\zeta_{\pm}$ becomes local holomorphic functions on $L_{\pm}$ , and one can check the definition does not depend on the choice of $\sigma$ . It is not hard to show using the defining equation of $\mathcal{N}^{0}$ that $\Omega_{\mathcal{N}^{0}}$ is a well-defined holomorphic volume form on $\mathcal{N}^{0}$ and is nowhere vanishing.

There is a natural $\mathbb{C}^{*}$ action on $\mathcal{N}^{0}$ given by

(4.90)

\lambda.(\zeta_{+},\zeta_{-})\equiv(\lambda^{-1}\zeta_{+},\lambda\zeta_{-}),\ \ \lambda\in\mathbb{C}^{*}.

and we denote by

(4.91)

\xi_{\mathcal{N}^{0}}\equiv\sqrt{-1}(-\zeta_{+}\partial_{\zeta_{+}}+\zeta_{-}\partial_{\zeta_{-}})

the corresponding holomorphic vector field (the choice of coefficients is made so that the real part of $\xi_{\mathcal{N}^{0}}$ is twice the real vector field generated by the induced $S^{1}$ action, as in (4.84)). One checks that

(4.92)

\xi_{\mathcal{N}^{0}}\lrcorner\ \Omega_{\mathcal{N}^{0}}=\Omega_{D}

Proposition 4.8.

There is a holomorphic embedding $\Phi:(\mathcal{M},\Omega)\rightarrow\mathcal{N}^{0}$ as a relatively compact open subset containing $\mathcal{P}$ , such that the following holds

(1)

$\Phi$ commutes with the projection maps to $D$ .
(2)

$\Phi^{*}\Omega_{\mathcal{N}^{0}}=\Omega.$
(3)

$d\Phi(\xi^{1,0})=\xi^{1,0}_{\mathcal{N}^{0}}$ . In particular, $\Phi$ maps $\mathcal{P}$ isomorphically onto $\mathcal{H}$ .

Remark 4.8.1.

From this we can say $D$ is indeed the GIT quotient of $\mathcal{N}_{0}$ , and we have a variation of GIT that leads to the birational map between $L_{+}^{-1}$ and $L_{-}$ .

Proof.

We define

(4.93)

\begin{cases}\mathcal{M}_{-}\equiv{\mathcal{M}^{*}}\setminus\pi^{-1}(H\times[0,\infty))\\ \mathcal{M}_{+}\equiv{\mathcal{M}^{*}}\setminus\pi^{-1}(H\times(\infty,0]).\end{cases}

On $\mathcal{M}_{-}$ we can trivialize the $U(1)$ connection $-\sqrt{-1}\Theta$ along the $z$ direction so that the $z$ component $\Theta_{z}$ vanishes identically. Denote by $\Theta|_{z}$ the restriction of $\Theta$ to the slice $D\times\{z\}$ for $z<0$ and to $(D\setminus H)\times\{z\}$ for $z\geq 0$ . From (4.33) we see that that curvature form of $-\sqrt{-1}\Theta|_{z}$ is given by $-\sqrt{-1}\partial_{z}\tilde{\omega}$ .

By Section 3.4, we have

(4.94)

\partial_{z}\tilde{\omega}|_{z=T_{-}}=k_{-}\omega_{D}+\epsilon_{T}

and

(4.95)

[\partial_{z}\tilde{\omega}]|_{z=T_{-}}=k_{-}[\omega_{D}]\in H^{2}(D;\mathbb{R}).

Since $b_{1}(D)=0$ , we may assume $\mathcal{M}|_{z=T_{-}}$ embeds into $L_{-}$ , as the unit circle bundle defined by another hermitian metric $\|\cdot\|_{\sim}^{2}$ which differs from the fixed metric by $\epsilon_{T}$ , and the connection 1-form $-\sqrt{-1}\Theta|_{T_{-}}$ agrees with the restriction of the Chern connection form. Denote by $\tilde{r}_{-}$ the norm function on $L_{-}$ corresponding to the new hermitian metric, then we have

(4.96)

\log\tilde{r}_{-}=\log r_{-}+\epsilon_{T}

Furthermore, we may extend $-\sqrt{-1}\Theta|_{T_{-}}$ naturally to the complement of the zero section ${\bf 0}_{L_{-}}$ in $L_{-}$ , via the fiberwise projection, and the resulting 1-form coincides with $\sqrt{-1}\tilde{r}_{-}^{-1}J_{-}d\tilde{r}_{-}$ , where $J_{-}$ denotes the complex structure on $L_{-}$ .

Now we define a map $\Phi_{-}:{\mathcal{M}^{*}}_{-}\rightarrow L_{-}\setminus{\bf 0}_{L_{-}}$ where ${\bf 0}_{L_{-}}$ denotes the zero section in $L_{-}$ . First at $z=T_{-}$ we define $\Phi_{-}$ to be the natural inclusion map as above, multiplied by $e^{A_{-}}$ for some constant $A_{-}$ to be determined later. Then using the trivialization of the $U(1)$ bundle ${\mathcal{M}^{*}}_{-}$ along the $z$ direction and the natural scaling map on $L_{-}$ , we extend the map to the whole ${\mathcal{M}^{*}}_{-}$ by setting

(4.97)

\tilde{r}_{-}=e^{A_{-}-\int_{T_{-}}^{z}h(u)du}

Then $\Phi_{-}$ clearly commutes with the projection maps to $D$ , so $\Phi_{-}^{*}\alpha=\alpha$ for any $1$ -form $\alpha$ which is a pull-back from $D$ . Since

(4.98)

\partial_{z}\Theta|_{z}=d_{D}^{c}h=-J_{D}d_{D}h

we have

(4.99)

\tilde{r}_{-}^{-1}\Phi_{-}^{*}d\tilde{r}_{-}=-hdz-\int_{T_{-}}^{z}du\wedge d_{D}h=-hdz-J_{D}(\Theta|_{z}-\Theta|_{T_{-}}),

noticing that $\Theta|_{z}-\Theta|_{T_{-}}$ is a 1-form pulled-back from $D$ . So

(4.100)

\tilde{r}_{-}^{-1}\Phi_{-}^{*}(d\tilde{r}_{-}+\sqrt{-1}J_{-}d\tilde{r}_{-})=-hdz-\sqrt{-1}\Theta|_{z}-\sqrt{-1}(\Theta|_{T_{-}}-\Theta|_{z})-J_{D}(\Theta|_{z}-\Theta_{T_{-}})

is a $(1,0)$ form on ${\mathcal{M}^{*}}_{-}$ .

Notice by definition locally

(4.101)

\tilde{r}_{-}^{2}=|\zeta_{-}|^{2}\cdot\|\sigma\|_{\sim}^{2}

(4.102)

\frac{d\zeta_{-}}{\zeta_{-}}=\frac{d\tilde{r}_{-}}{\tilde{r}_{-}}+\sqrt{-1}J_{-}\frac{d\tilde{r}_{-}}{\tilde{r}_{-}}+\partial_{D}\log|\sigma|^{2}

Therefore we obtain

(4.103)

\Phi_{-}^{*}\Omega_{L_{-}}=\Omega

where

(4.104)

\Omega_{L_{-}}\equiv-\sqrt{-1}\frac{d\zeta_{-}}{\zeta_{-}}\wedge\Omega_{D}

is a natural holomorphic volume form on $L_{-}\setminus{\bf 0}_{L_{-}}$ . In particular $\Phi_{-}$ is a holomorphic embedding. Also, we have

(4.105)

d\Phi_{-}(\xi^{1,0})=\sqrt{-1}\zeta_{-}\partial_{\zeta_{-}}

is the natural holomorphic vector field on $L_{-}$ .

Since $h$ is positive we see that the image of $\Phi_{-}$ is bounded in $L_{-}$ . Since $\mathcal{M}\setminus\mathcal{M}_{-}$ is of complex codimension one, by the removable singularity theorem for bounded holomorphic functions, $\Phi_{-}$ extends to a holomorphic map on the entire $\mathcal{M}$ .

Similarly we get a holomorphic embedding

(4.106)

\Phi_{+}:\mathcal{M}_{+}\rightarrow L_{+}

with

(4.107)

\tilde{r}_{+}=e^{A_{+}-\int_{z}^{T_{+}}h(u)du}

for a constant $A_{+}$ to be determined. Again $\Phi_{+}$ extends to a holomorphic map on $\mathcal{M}$ .

Together we obtain

(4.108)

\Phi\equiv(\Phi_{+},\Phi_{-}):\mathcal{M}\rightarrow L_{+}\oplus L_{-}

which is an embedding on $\mathcal{M}\setminus\mathcal{P}$ . It commutes with projections maps to $D$ and satisfies

(4.109)

d\Phi(\xi^{1,0})=\sqrt{-1}(\zeta_{-}\partial_{\zeta_{-}}-\zeta_{+}\partial_{\zeta_{+}}).

Now we show that with appropriate choice of $A_{\pm}$ , $\Phi$ maps $\mathcal{M}$ into $\mathcal{N}^{0}$ . First we notice that by (4.109),

(4.110)

\det\Phi\equiv\Phi_{+}\otimes\Phi_{-}:\mathcal{M}\setminus(H\times(-\infty,\infty))\rightarrow L_{+}\otimes L_{-}

has image lying on a non-zero holomorphic section, say $\tilde{S}$ , of $L^{\otimes k}=L_{+}\otimes L_{-}$ over $D\setminus H$ . By definition since $h$ is positive we know the the image of $\Phi$ is bounded in $L_{+}\oplus L_{-}$ , with respect to the norm $\tilde{r}_{\pm}$ , so $\tilde{S}$ is a bounded section of $L^{\otimes k}$ with respect to the norm $\tilde{r}\equiv\tilde{r}_{+}\otimes\tilde{r}_{-}$ , hence again by removable singularity theorem for bounded holomorphic functions it extends to a holomorphic section on the entire $D$ . By our assumption that $[H]$ is isomorphic to $L$ , we see $H$ is exactly the zero locus of $\tilde{S}$ , so there is a constant $C$ such that

(4.111)

\tilde{S}=C\cdot S_{H}.

Multiplying $\Phi_{-}$ by an element in $S^{1}$ we may assume $C$ is a positive real number. Now

(4.112)

\log C=\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|{\tilde{S}}\|\omega_{D}^{n-1}-\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|\omega_{D}^{n-1}

The second term is a constant independent of $T$ . For the first term, by definition we have

(4.113)

-\log\|\tilde{S}\|=\int_{T_{-}}^{T_{+}}hdz-(A_{-}+A_{+})+\epsilon_{T}.

By (4.14)

(4.114)		$\displaystyle\int_{T_{-}}^{T_{+}}\int h\omega_{D}^{n-1}$	$\displaystyle=$	$\displaystyle\int_{T_{-}}^{T_{+}}T^{2-n}\int_{D}(T\omega_{D}+\psi)^{n-1}dz+T^{-1}\underline{B}_{T}$
(4.115)			$\displaystyle=$	$\displaystyle\frac{1}{n}\int_{D}\omega_{D}^{n-1}(\frac{1}{k_{-}}-\frac{1}{k_{+}})(T^{2}-1)+T^{-1}\underline{B}_{T}$

So we get that

(4.116)

-\log C=\frac{1}{n}(\frac{1}{k_{-}}-\frac{1}{k_{+}})(T^{2}-1)-(A_{-}+A_{+})+\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|+T^{-1}\underline{B}_{T}

Setting $C=1$ gives one condition on $A_{-}$ and $A_{+}$ . For our later purposes we shall need additionally that

(4.117)

k_{-}A_{-}=k_{+}A_{+}

Together these determine $A_{-}$ and $A_{+}$ as

(4.118)

A_{-}\equiv\frac{1}{nk_{-}}(T^{2}-1)-\frac{-k_{+}}{2(k_{-}-k_{+})}\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|+T^{-1}\underline{B}_{T}

(4.119)

A_{+}\equiv\frac{1}{-nk_{+}}(T^{2}-1)-\frac{k_{-}}{2(k_{-}-k_{+})}\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|+T^{-1}\underline{B}_{T}

Then we can make $\Phi$ maps $\mathcal{M}$ into $\mathcal{N}^{0}$ .

It is easy to check that $\Phi$ satisfies (1), (2), (3) in the statement of the Proposition. It then follows from (2) that $\Phi$ is a holomorphic embedding also across $\mathcal{P}$ . This finishes the proof of Proposition.

∎

Remark 4.8.2.

In the case $b_{1}(D)\neq 0$ , from the proof we can make the same conclusion except the holomorphic line bundles $L_{+}$ and $L_{-}$ can not be prescribed as isomorphic to the powers on the given holomorphic line bundle $L$ . Instead, as can be seen in the above proof, they are determined by the restriction of $\partial_{z}\tilde{\omega}(z)$ on the two ends. However, as pointed in Remark 4.3.2, we always have $L_{+}=L^{-\otimes k_{+}}\otimes\mathcal{F}$ and $L_{-}=L^{\otimes k_{-}}\otimes\mathcal{F}^{-1}$ for some holomorphic line bundle $\mathcal{F}$ on $D$ with $c_{1}(\mathcal{F})=0$ . In particular the tensor product $L_{+}\otimes L_{-}$ is always isomorphic to $L^{k}$ . The freedom of $\mathcal{F}$ corresponds exactly to the choice of the connection 1-form $\Theta$ in the construction of $\mathcal{M}^{*}$ .

For our purpose later, we list a few more results here. First we shall need to compare the function $z$ with the norm $r_{-}$ and $r_{+}$ near each end. Given $C>0$ fixed, then by (4.16) we have

(4.120)

\begin{cases}-\log r_{-}=\frac{1}{nk_{-}}T^{2-n}(T+k_{-}z)^{n}-A_{-}+\epsilon_{T}+\epsilon(z),\ \ z\leq-C;\\ -\log r_{+}=-\frac{1}{nk_{+}}T^{2-n}(T+k_{+}z)^{n}-A_{+}+\epsilon_{T}+\epsilon(z),\ \ z\geq C.\end{cases}

by noticing that for example

(4.121)

\int_{T_{-}}^{z}\epsilon(z)dz=\epsilon_{T}+\epsilon(z),z\leq-C.

So we have

(4.122)

\begin{cases}(T+k_{-}z)^{n}=T^{n-2}nk_{-}(A_{-}-\log r_{-}+\epsilon_{T}+\epsilon(z))\\ (T+k_{+}z)^{n}=-T^{n-2}nk_{+}(A_{+}-\log r_{-}+\epsilon_{T}+\epsilon(z))\end{cases}

For our analysis later we also give a description of the behavior of the metric $\omega$ when we restrict to the region $|z|\geq 1$ . From the asymptotics of $\tilde{\omega}$ and $h$ we know the metric is asymptotic to the Calabi model space in Section 2.2. Locally on $D$ we fix holomorphic coordinates $\{w_{1},\cdots,w_{n-1}\}$ and choose a holomorphic trivialization of $L$ as before, then we obtain fiber holomorphic coordinates $\zeta_{\pm}$ on $L_{\pm}$ . Denote

(4.123)

\omega_{\pm,cyl}\equiv\sum_{i\geq 1}\sqrt{-1}dw_{i}\wedge d\bar{w}_{i}+\frac{\sqrt{-1}d\zeta_{\pm}\wedge d\bar{\zeta}_{\pm}}{|\zeta_{\pm}|^{2}}

the local cylindrical type metrics on $L_{\pm}$ respectively. Then we have

Lemma 4.9.

On $|z|\geq 1$ , we have

(4.124)

C^{-1}T^{\frac{(n-2)(1-n)}{n}}(T+k_{\pm}z)^{1-n}\omega_{\pm,cyl}\leq\omega\leq CT^{\frac{2-n}{n}}(T+k_{\pm}z)\cdot\omega_{\pm,cyl}

and for all $k\geq 1$ , there exists $m_{k},C_{k}$ such that

(4.125)

|\nabla^{k}_{\omega_{\pm,cyl}}\omega|_{\omega_{\pm,cyl}}\leq C_{k}(T^{\frac{2-n}{n}}(T+k_{\pm}z))^{m_{k}}.

Proof.

Consider the case $z\leq-1$ . Since

(4.126)

\frac{d\zeta_{-}}{\zeta_{-}}=\frac{dr_{-}}{r_{-}}+\sqrt{-1}J\frac{dr_{-}}{r_{-}}=-hdz+\epsilon_{T}-\sqrt{-1}Jhdz

The result then easily follows from the asymptotics of $h$ (4.16) and $\tilde{\omega}$ (3.349). ∎

Finally we need to understand the boundary of the shape of the level set $r_{\pm}=C$ under the projection to $D\times\mathbb{R}$ , for a fixed $C>0$ and for $T$ large. First we have the formula

Lemma 4.10.

We have

(4.127)

A_{-}-\int_{T_{-}}^{0}h(u)du=\frac{1}{2}\log{\|S_{H}\|}+B_{T}

(4.128)

A_{+}+\int_{T_{+}}^{0}h(u)du=\frac{1}{2}\log{\|S_{H}\|}+B_{T}

Proof.

We denote

(4.129)

\hat{h}_{-}=A_{-}-\int_{T_{-}}^{0}h(u)du-\frac{1}{2}\log\|S_{H}\|.

By the Poincaré-Lelong equation we have

(4.130)

d_{D}d_{D}^{c}\log{\|S_{H}\|}^{2}=4\pi\delta_{H}-(k_{-}-k_{+})\omega_{D},

where $\delta_{H}$ denotes the current of integration along $H$ . By directly taking derivatives and use (2.13) we obtain that outside $H$ ,

(4.131)

d_{D}d_{D}^{c}(\int_{0}^{T_{-}}h(z)dz)=\int_{0}^{T_{-}}d_{D}d_{D}^{c}h(z)dz=\int_{0}^{T_{-}}-\partial_{z}^{2}\tilde{\omega}(z)dz=-\partial_{z}\tilde{\omega}|_{z=T_{-}}+\partial_{z}\tilde{\omega}|_{z=0}

By (3.349) and (3.381), the right hand side is given by $-\frac{1}{2}(k_{-}-k_{+})\omega_{D}+\epsilon_{T}$ . Now using the asymptotics of $h$ near $P$ in (4.17), one sees that $\hat{h}_{-}$ is bounded near $H$ . So the following current equation holds globally on $D$

(4.132)

d_{D}d_{D}^{c}\hat{h}_{-}=\epsilon_{T}

Now

(4.133)

\int_{D}\hat{h}_{-}\omega_{D}^{n-1}=A_{-}\int_{D}\omega_{D}^{n-1}+\int_{D}\int_{0}^{T_{-}}h\omega_{D}^{n-1}dz=B_{T}

So by standard elliptic regularity we get the conclusion for $\hat{h}_{-}$ . The proof for the other equation is similar. ∎

Since for $|z|\leq 1$ we have $h(z)=T+\frac{1}{2r}+O^{\prime}(r)+O(T^{-1})$ , we easily see that in a fixed distance (with respect to $\omega_{D}$ ) away from $H$ , $r_{\pm}\leq C$ is equivalent to $B_{T}\cdot T^{-1}\mp z\geq 0$ . Now we fix a point in $H$ and as before consider the coordinate chart $(y,\bar{y},w_{2}^{\prime},\cdots,\bar{w}_{n-1}^{\prime})$ on $D$ centered at this point. Then we have

Proposition 4.11.

In this chart we have

(4.134)		$\displaystyle\log r_{-}$	$\displaystyle=-Tz+\frac{1}{2}\log(r-z)+B_{T},$
(4.135)		$\displaystyle\log r_{+}$	$\displaystyle=Tz+\frac{1}{2}\log(r+z)+B_{T}.$

Proof.

By the previous Lemma,

(4.136)

A_{-}-\int_{T_{-}}^{z}h(u)=\frac{1}{2}\log{\|S_{H}\|}+B_{T}+\int_{z}^{0}h(u)du

When $|z|\leq 1$ , if we are in the above chart, then

(4.137)

\int_{0}^{z}h(u)du=B_{T}+Tz+\frac{1}{2}(\log(r+z)-\log|y|)

Since $\log{\|S_{H}\|}=\log|y|+B_{T}$ , it follows that

(4.138)

\log r_{-}=B_{T}-Tz+\frac{1}{2}\log(r-z)

Similarly we get the estimate for $\log r_{+}$ .

∎

Corollary 4.11.1.

The following hold:

(1)

Let $C>0$ be fixed, then for $T$ large, $r_{-}\leq C$ implies $z\geq-\frac{3}{4}T^{-1}\log T$ .
(2)

Let $c>0$ be fixed. Then for $T$ large if $r\leq cT^{-1}\log T$ for some $c<1/2$ , then

$\log r_{-}\leq-\frac{1}{2}(\frac{1}{2}-c)\log T.$
(3)

Let $C\geq 1$ be fixed, then for $T$ large, $z\geq-C$ implies $\log r_{-}\leq(C+1)T$

Proof.

The first two items are easy consequences of the previous Lemma. For the last item we simply notice that for $C\geq 1$ ,

(4.139)

\int_{-C}^{-1}h(u)du=\int_{-C}^{-1}(T^{2-n}(T+k_{-}u)^{n-1}+\epsilon(u))du\leq CT.

∎

4.2.2. Kähler potentials

We look for an $S^{1}$ invariant function $\phi$ on $\mathcal{M}$ satisfying the equation

(4.140)

T\pi^{*}\omega_{D}+dd^{c}\phi=T^{\frac{n-2}{n}}\omega

We write

(4.141)

d\phi=d_{D}\phi+\phi_{z}dz

where as before $d_{D}\phi$ is the differential along $D$ direction and $\phi_{z}=\partial_{z}\phi$ is the derivative along $z$ direction. Then

(4.142)

d^{c}\phi=d^{c}_{D}\phi+\phi_{z}h^{-1}\Theta,

and

(4.143)

dd^{c}\phi=d_{D}d_{D}^{c}\phi+dz\wedge(d^{c}_{D}\phi_{z})+d(\phi_{z}h^{-1})\wedge\Theta+\phi_{z}h^{-1}(\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h)

Since

(4.144)

T^{\frac{n-2}{n}}\omega=\pi^{*}\tilde{\omega}+dz\wedge\Theta,

we see (4.140) is equivalent to the system of equations

(4.145)

\begin{cases}\tilde{\omega}=T\omega_{D}+d_{D}d^{c}_{D}\phi+\phi_{z}h^{-1}\partial_{z}\tilde{\omega}\\ d_{D}^{c}\phi_{z}-\phi_{z}h^{-1}d_{D}^{c}h=0\\ d(\phi_{z}h^{-1})=dz.\end{cases}

To solve these (apparently overdetermined) equations, we first notice that the last equation in (4.145) is equivalent to

(4.146)

\phi_{z}h^{-1}=z+C

for a constant $C$ . So we obtain ²² 2 In the case when $n=2$ for the classical Gibbons-Hawking ansatz this formula was derived by the authors together with Hans-Joachim Hein in the office of the first author at Stony Brook in the Fall of 2017.

(4.147)

\phi(z)=\int_{z_{0}}^{z}(u+C)hdu+\phi(z_{0})

for a function $\phi(z_{0})$ on $D$ .

The second equation of (4.145) then holds automatically, and the first equation also follows after taking $\partial_{z}$ . So in order for $\phi$ defined in (4.147) to satisfy (4.145), it suffices that at a fixed $z=T_{+}$ the following holds

(4.148)

T\omega_{D}+d_{D}d^{c}_{D}\phi=\tilde{\omega}-(T_{+}+C)\partial_{z}\tilde{\omega}

Comparing the cohomology class of both sides yields that $C$ must be zero. Then we can solve $\phi(T_{+})$ uniquely up to addition of a constant. After fixing a choice of $\phi(T_{+})$ we may define $\phi$ by

(4.149)

\phi(z)=\int_{T_{+}}^{z}uhdu+\phi(T_{+})

and we can view it as either a function on $Q_{T}$ or an $S^{1}$ invariant function on $\mathcal{M}$ .

Proposition 4.12.

The function $\phi$ is smooth on ${\mathcal{M}^{*}}$ , and $C^{3,\alpha}$ on $\mathcal{M}$ (in the smooth topology as defined in Section 4.1), and satisfies (4.140).

Remark 4.12.1.

The regularity is indeed $C^{4,\alpha}$ in local holomorphic coordinates.

Proof.

By definition $\phi$ is smooth on $Q_{T}\setminus H\times(-\infty,0]$ . Using (4.17) it is easy to see that $\phi$ extends to a continuous function on $Q_{T}$ . Hence for all fixed $z$ , the following equation holds in the sense of currents on $D$

(4.150)

T\omega_{D}+d_{D}d_{D}^{c}\phi(z)=\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z).

Elliptic regularity then implies that $\phi$ is smooth on each slice $\{z\}\times D$ for $z\neq 0$ . Now for $z\leq 0$ we can write

(4.151)

\phi(z)=\int_{T_{-}}^{z}uhdu+\phi(T_{-}).

We then see that $\phi$ is indeed smooth on $Q_{T}\setminus P$ . Over the $S^{1}$ fibration $\mathcal{M}$ , we know $\phi$ is globally continuous, and it is smooth and satisfies the equation (4.140) on ${\mathcal{M}^{*}}$ . Now again by standard theory on pluri-subharmonic functions we conclude the current equation holds on $\mathcal{M}$ . Since we know $\omega$ is $C^{2,\alpha}$ in local holomorphic coordinates on $\mathcal{M}$ , elliptic regularity gives that $\phi$ is in $C^{4,\alpha}$ in local holomorphic coordinates. This implies that $\phi$ is $C^{3,\alpha}$ in the smooth topology we defined, since we know the holomorphic coordinate functions are $C^{3,\alpha}$ . ∎

Remark 4.12.2.

As a by-product we can also recover the formula of the Calabi model metric in terms of Kähler potentials as mentioned in Section 2.2. In this case as in (2.30) we take $\tilde{\omega}=z\omega_{D}$ and $h=z^{n-1}$ . Then we can write

(4.152)

\tilde{\omega}=dd^{c}\phi

with

(4.153)

\phi=\int_{0}^{z}u^{n}du=\frac{1}{n+1}z^{n+1}

To match with the formula for Calabi ansatz in (2.32), we notice that $z^{n+1}=(-\log|\xi|)^{2}$ , and there is a factor of $\frac{n}{2}$ due to the normalization of the Calabi-Yau equation and that $dd^{c}=2\sqrt{-1}\partial\bar{\partial}$ .

Remark 4.12.3.

Notice the argument above does not essentially require the compactness of $D$ , except to solve the equation (4.150) on one slice. Using similar idea can get the expression of the Taub-NUT metric on $\mathbb{C}^{2}$ in terms of Kähler potentials, as mentioned in Section 2.3. Here we take $D$ to be $\mathbb{C}$ with the standard flat structure, and

(4.154)

\tilde{\omega}(z)=\frac{\sqrt{-1}}{2}Vdy\wedge d\bar{y};\ \ \ \ h=V,

with

(4.155)

V=\frac{1}{2r}+T.

Suppose we want to find $\phi$ with

(4.156)

\omega=dd^{c}\phi,

then we first have

(4.157)

\phi(z)-\phi(0)=\int_{0}^{z}(\frac{1}{2r}+T)du=\frac{1}{2}r-\frac{1}{2}|y|+\frac{T}{2}z^{2}

The equation (4.150) for $z=0$ becomes

(4.158)

4\partial_{y}\partial_{\bar{y}}\phi(0)=\tilde{\omega}(0)=\frac{1}{2|y|}+T

and a solution is given by

(4.159)

\phi(0)=\frac{1}{2}|y|+\frac{T}{4}|y|^{2}

So we get

(4.160)

\phi=\frac{1}{2}r+\frac{T}{2}z^{2}+\frac{T}{4}|y|^{2}.

In terms of the $u_{1},u_{2}$ coordinates we get

(4.161)

\phi=\frac{1}{4}(|u_{1}|^{2}+|u_{2}|^{2})+\frac{T}{8}(|u_{1}|^{4}+|u_{2}|^{4}).

This agrees with formula (7.61) up to a constant $2$ , again caused by the fact that $dd^{c}=2\sqrt{-1}\partial\bar{\partial}$ .

Notice from the above discussion we know for each fixed $z$ , $\phi(z)$ is uniquely determined up to a constant on $D$ by the equation

(4.162)

T\omega_{D}+d_{D}d^{c}_{D}\phi(z)=\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z),

and the integration formula (4.149) exactly gives a coherent way of fixing all the constants for each $z$ , so the overall freedom in only up to a global constant. ³³ 3 maybe more geometric explanation if we have time

Notice by (3.349) we have for $z\gg 1$ ,

(4.163)

\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z)-T\omega_{D}=\psi(z)-z\partial_{z}\psi(z)=\epsilon(z)

Standard elliptic estimate allows us to find a solution $\phi(T_{+})$ which is $\epsilon_{T}$ . By (4.16) we obtain that for $z\geq C$

(4.164)

\phi(z)=C_{+}+T^{2-n}k_{+}^{-2}[\frac{(k_{+}z+T)^{n+1}}{n+1}-\frac{T(k_{+}z+T)^{n}}{n}]

where

(4.165)

C_{+}=\epsilon_{T}+\epsilon(z)-T^{2-n}k_{+}^{-2}[\frac{1}{n+1}T^{\frac{(n+1)(n-2)}{n}}-\frac{1}{n}T^{n-2}]

For the other end $z\leq-C$ , similarly we have

(4.166)

\phi(z)-\phi(T_{-})=C_{-}+T^{2-n}k_{-}^{-2}[\frac{(k_{-}z+T)^{n+1}}{n+1}-\frac{T(k_{-}z+T)^{n}}{n}]

where

(4.167)

C_{-}=\epsilon_{T}+\epsilon(z)+T^{2-n}k_{-}^{-2}[\frac{1}{n+1}T^{\frac{(n+1)(n-2)}{n}}-\frac{1}{n}T^{n-2}]

To understand $\phi(T_{-})$ we need the following

Lemma 4.13.

We have

(4.168)

\phi(T_{-})=\epsilon_{T}+T^{-1}\underline{B}_{T}

Proof.

We have $\phi(T_{-})=\phi(T_{+})-\Psi,$ where

(4.169)

\Psi=\int_{T_{-}}^{T_{+}}zhdz.

Away from $H$ we have

(4.170)

d_{D}d_{D}^{c}\Psi=\int_{T_{-}}^{T_{+}}zd_{D}d_{D}^{c}hdz=-\int_{T_{-}}^{T_{+}}z\partial_{z}^{2}\tilde{\omega}dz

Integration by parts we get

(4.171)

d_{D}d_{D}^{c}\Psi=(-z\partial_{z}\tilde{\omega}+\tilde{\omega})|^{T_{+}}_{T_{-}}=\epsilon_{T}

Notice since there is a factor $z$ in the integrand we do not get residue term at $z=0$ . Notice $\Psi$ is continuous on $D$ , and the right hand side is smooth on $D$ , so elliptic regularity implies that $\Psi$ is indeed smooth on $D$ , and the equation holds globally on $D$ .

On the other hand, we have

(4.172)

\int_{D}\Psi\omega_{D}^{n-1}=\int_{T_{-}}^{T_{+}}z\int_{D}h\omega_{D}^{n-1}dz

Using (4.14)

	$\displaystyle\frac{\int_{D}\Psi\omega_{D}^{n-1}}{\int_{D}\omega_{D}^{n-1}}=$	$\displaystyle T^{2-n}k_{+}^{-2}[\frac{(k_{+}z+T)^{n+1}}{n+1}-\frac{T(k_{+}z+T)^{n}}{n}]$
(4.173)			$\displaystyle-T^{2-n}k_{-}^{-2}[\frac{(k_{-}z+T)^{n+1}}{n+1}-\frac{T(k_{-}z+T)^{n}}{n}]+T^{-1}\underline{B}_{T},$

where we used the definition of $T_{-}$ and $T_{+}$ . (4.171) and (4.173) together yield the conclusion. ∎

Now we investigate (4.166).

(4.174)

\phi(z)-\phi(T_{-})=C_{-}+T^{2-n}k_{-}^{-2}[\frac{(k_{-}z+T)^{n+1}}{n+1}-\frac{T(k_{-}z+T)^{n}}{n}]+\epsilon_{T}+O(T^{-1})

We first notice that by (4.120)

(4.175)

-T^{3-n}k_{-}^{-2}\frac{(k_{-}z+T)^{n}}{n}=\frac{T}{k_{-}}(\log r_{-}-A_{-}+\epsilon_{T}+\epsilon(z))+\epsilon_{T}

We may also write by definition

(4.176)

\omega_{D}=-\frac{1}{k_{-}}dd^{c}\log r_{-}

So when $z\leq-C$ , we have

(4.177)

T\pi^{*}\omega_{D}+dd^{c}\phi=T^{\frac{n-2}{n}}dd^{c}\phi_{-},

with

(4.178)

\phi_{-}\equiv\frac{1}{n+1}n^{\frac{n+1}{n}}k_{-}^{-\frac{n-1}{n}}(A_{-}+\epsilon_{T}+\epsilon(z)-\log r_{-})^{\frac{n+1}{n}}-T^{\frac{2}{n}}k_{-}^{-1}A_{-}+\epsilon_{T}+T^{\frac{2}{n}}\epsilon(z)

Similarly for $z\geq C$ , we have

(4.179)

T\pi^{*}\omega_{D}+dd^{c}\phi=dd^{c}\phi_{+},

with

(4.180)

\phi_{+}\equiv\frac{1}{n+1}n^{\frac{n+1}{n}}(-k_{+})^{-\frac{n-1}{n}}(A_{+}+\epsilon_{T}+\epsilon(z)-\log r_{+})^{\frac{n+1}{n}}-T^{\frac{2}{n}}k_{+}^{-1}A_{+}+\epsilon_{T}+T^{\frac{2}{n}}\epsilon(z).

4.3. Geometries at regularity scales

In this subsection, we will take a closer look at the Riemannian geometric behavior of the family of incomplete Kähler metrics $(\mathcal{M}_{T},\omega_{T})$ constructed in Section 4.1 as $T\rightarrow\infty$ . For clarity we now re-install the parameter $T$ throughout the rest of this section.

It is easy to see that as the parameter $T\to+\infty$ , the curvatures are unbounded around the singular set $\mathcal{P}\subset\mathcal{M}_{T}$ such that the standard uniform elliptic estimates just legitimately fail. Instead, we will define some appropriate weighted Hölder spaces and establish uniformly weighted a priori estimates, which will be done in Section 4.4. Geometrically, the weighted elliptic estimate that we pursue is intimately connected with the effective regularity at definite scales of the metrics $\omega_{T}$ in various pieces of $\mathcal{M}_{T}$ . More rigorously, we need the following notion.

Definition 4.14 (Local regularity).

Let $(M^{n},g,p)$ be a Riemannian manifold and $p\in M^{n}$ . Given $r>0$ , $\epsilon>0$ , $k\in\mathbb{N}$ , $\alpha\in(0,1)$ , we say $(M^{n},g,p)$ is $(r,k+\alpha,\epsilon)$ -regular at $p$ if the metric $g$ is at least $C^{k+\alpha}$ in $B_{2r}(p)$ and satisfies the following property: let $(\widetilde{B_{2r}(p)},\tilde{p})$ be the Riemannian universal cover of $B_{2r}(p)$ , then $B_{r}(\tilde{p})$ is diffeomorphic to a disc $\mathbb{D}^{n}\subset\mathbb{R}^{n}$ such that $g$ in coordinates satisfies

(4.181)

|g_{ij}-\delta_{ij}|_{C^{0}(B_{r}(\tilde{p}))}+\sum\limits_{m=1}^{k}r^{m}\cdot|\nabla^{m}g_{ij}|_{C^{0}(B_{r}(\tilde{p}))}+r^{k+\alpha}[g_{ij}]_{C^{k,\alpha}(B_{r}(\tilde{p}))}<\epsilon.

Definition 4.15 ( $C^{k,\alpha}$ -regularity scale).

Let $(M^{n},g)$ be a Riemannian manifold with a $C^{k,\alpha}$ -Riemannian metric $g$ . The $C^{k,\alpha}$ -regularity scale at $p$ , denoted by $r_{k,\alpha}(p)$ , is defined as the supremum of all $r>0$ such that $M^{n}$ is $(r,k+\alpha,10^{-6})$ -regular at $p$ .

Intuitively, the $C^{k,\alpha}$ -regularity scale is the maximal zooming-in scale at which the nontrivial $C^{k,\alpha}$ -geometry is uniformly bounded on the local universal cover, which maximally captures the bounded covering $C^{k,\alpha}$ -geometry.

Example 4.16.

If $g$ is a $C^{k,\alpha}$ -metric on $M^{n}$ , then for any $p\in M^{n}$ , we have $r_{k,\alpha}(p)>0$ . Here the size of $r_{k,\alpha}(p)$ depends on $p$ .

Example 4.17.

Let $(M^{n},g)$ satisfy $|\Rm_{g}|\leq 1$ in $B_{2}(p)$ , then the following holds:

(1)

there exists a dimensional constant $r_{0}(n)>0$ such that $r_{1,\alpha}(x)\geq r_{0}(n)>0$ for all $x\in B_{1}(p)$ and $\alpha\in(0,1)$ . Moreover, $r_{1,\alpha}(p)\geq r_{0}(n)\cdot r_{|\Rm|}(p)>0$ , where

(4.182) $r_{|\Rm|}(p)\equiv\sup\Big\{r>0\Big||\Rm|_{C^{0}(B_{r}(p))}\leq r^{-2}\Big\}$

denotes the curvature scale at $p$ .
(2)

In particular, if $\Rm_{g}\equiv 0$ on a complete manifold $M^{n}$ , then $r_{k,\alpha}(x)=+\infty$ for all $x\in M^{n}$ , $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ .

The goal of this subsection is to study the $(k,\alpha)$ -regularity scale at every point for appropriate $k,\alpha$ . Since the Kähler metrics $\omega_{T}$ constructed in Section 4.1 are fairly explicit, so for every $\bm{x}\in\mathcal{M}_{T}$ we will explicitly determine a canonical scale $\mathfrak{s}(\bm{x})$ which is convenient for calculations and uniformly proportional to the $(k,\alpha)$ -regularity scale $r_{k,\alpha}(\bm{x})$ at $\bm{x}$ , i.e.

(4.183)

\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x})\leq\mathfrak{s}(\bm{x})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}),

for some uniform constants $\underline{v}_{0}$ and $\bar{v}_{0}>0$ which are independent of $T\gg 1$ . For convenience, $\mathfrak{s}(\bm{x})$ will be called the regularity scale.

Remark 4.17.1.

Without loss of generality, in the discussion below, we always assume that the curvatures of $\omega_{D}$ is not identically zero. Otherwise, one can work at even larger scale for some regions, but we do not need that for our purpose.

Before the technical computations, it is helpful to present the scenario of geometric transformations on $\mathcal{M}_{T}$ from the singular set $\mathcal{P}$ to the boundary $\partial\mathcal{M}_{T}$ . First, as $T\to+\infty$ , curvatures blow up if the reference point $\bm{x}$ is located around $\mathcal{P}$ so that we will rescale the metric $\omega_{T}$ giving rise to a product bubble limit $\mathbb{C}_{TN,\varrho}^{2}\times\mathbb{C}^{n-2}$ , where $\mathbb{C}_{TN,\varrho}^{2}$ is the Taub-NUT space (c.f. Section 2.3) for some $\varrho>0$ . This is a deepest bubble (rescaling limit) in our context. When the distance from $\bm{x}$ to $\mathcal{P}$ is increasing, the length of $S^{1}$ -fiber at the infinity of the Taub-NUT space $\mathbb{C}_{TN,\varrho}^{2}\times\mathbb{C}^{n-2}$ is decreasing which corresponds to $\varrho$ is increasing. The next level of bubble corresponds to $\varrho\rightarrow\infty$ , or equivalently, this amounts to getting the tangent cone at infinity of the product $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ , which is $\mathbb{R}^{2n-1}\equiv\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ . This is of codimension- $1$ collapse, with locally uniformly bounded curvature away from $\{0^{3}\}\times\mathbb{C}^{n-2}$ . When $\bm{x}$ is getting further away from $\mathcal{P}$ , the size of $D$ will be shrinking such that the next level of bubble is $D\times\mathbb{R}$ . This is again a codimension- $1$ collapse, with locally uniformly bounded curvature away $P=H\times\{0\}\subset D\times\mathbb{R}$ . Finally, as $\bm{x}$ moves close to the boundary $\partial\mathcal{M}_{T}$ , the metrics will converge to the incomplete Calabi model metrics $\mathcal{C}^{n}_{-}$ and $\mathcal{C}^{n}_{+}$ , which corresponds to applying the construction in Section 2.2 to the line bundle $L^{k_{-}}$ and $L^{k_{+}}$ over $D$ .

Now we are ready to make precise subdivision for $\mathcal{M}_{T}$ and analyze different rescaling geometries (see Figure 4.1). Let $H$ be a divisor of $D$ such that the singular set $P=H\times\{0\}$ is at the slice $z=0$ of the cylinder $Q\equiv D\times\mathbb{R}$ . Denote by $r(\bm{x})$ the distance from $\pi(\bm{x})$ to $P$ with respect to the product metric $g_{Q}=g_{D}+dz^{2}$ on the base $Q$ .

Region $\bf{I}_{1}$ :

This region consists of the points $\bm{x}$ satisfying

(4.184)

r(\bm{x})\leq T^{-1}.

In other words, this region consists of points close to the divisor $P=H\times\{0\}\subset D\times\{0\}$ which is the singular locus of the $S^{1}$ -fibration.

Region $\bf{I}_{2}$ :

A point $\bm{x}$ in this region satisfies

(4.185)

\frac{T^{-1}}{2}\leq r(\bm{x})\leq 1.

So this region contains the points not close, but not too far from the divisor $P$ .

Region $\bf{I}_{3}$ :

This region consists of the points far from the divisor $P$ such that each $\bm{x}$ satisfies the condition

(4.186)

\displaystyle r(\bm{x})\geq\frac{1}{2},\quad T_{-}\leq z(\bm{x})\leq T_{+}.

Notice that the above regions completely cover the neck $\mathcal{M}_{T}$ such that each overlapping region has the same geometric behavior with the adjacent regions in the above subdivision. So we will just ignore these overlaps in the following discussions.

Figure 4.1. Subdivision of

\mathcal{M}_{T}

into various regions

Under the above subdivision of $\bf{I}_{1}$ , $\bf{I}_{2}$ and $\bf{I}_{3}$ , we will rather explicitly determine the corresponding $(k,\alpha)$ -regularity scales with respect to the metric

(4.187)

\omega_{T}\equiv T^{\frac{2-n}{n}}(\pi^{*}\tilde{\omega}+dz\wedge\Theta).

Region $\bf{I}_{1}$ (the deepest bubble):

For each point $\bm{x}$ in this region, we choose

(4.188)

\mathfrak{s}(\bm{x})=T^{\frac{1-n}{n}}.

As in (2.52), let us denote by

(4.189)

\displaystyle\begin{cases}\omega_{TN,1}=(\frac{1}{2r}+1)\cdot\frac{\sqrt{-1}}{2}\cdot dy\wedge d\bar{y}+dz\wedge\Theta_{0},\\ \Omega_{TN,1}=\sqrt{-1}((\frac{1}{2r}+1)dz+\Theta_{0})\wedge dy\end{cases}

the Kähler form and the holomorphic form of the Taub-NUT space $\mathbb{C}_{TN}^{2}$ whose $S^{1}$ -fiber at infinity has length equal to $1$ .

In the following, we will carry out explicit calculations to prove that under the rescaled metric

(4.190)

\tilde{g}_{T}=(\mathfrak{s}(\bm{x}))^{-2}g_{T}=T^{\frac{2n-2}{n}}g_{T},

we have the pointed convergence

(4.191)

(\mathcal{M}_{T},\tilde{g}_{T},\bm{x})\xrightarrow{C^{2,\alpha}}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\omega_{TN,1}\oplus g_{\mathbb{C}^{n-2}},(\bm{0}^{2},0^{n-2}))

in the pointed $C^{2,\alpha}$ -topology, where $\bm{0}^{2}$ is the origin of the Taub-NUT space $(\mathbb{C}_{TN}^{2},\omega_{TN,1})$ . Moreover, the rescaled holomorphic volume form $T^{n-1}\cdot\Omega_{T}$ converges to $\Omega_{TN,1}$ in the $C^{2,\alpha}$ -topology, where $\Omega_{TN,1}$ is the holomorphic volume form of $(\mathbb{C}_{TN}^{2},{\omega}_{TN,1})$ (c.f. Section 2.3). This implies that

(4.192)

\underline{v}_{0}\cdot r_{2,\alpha}(\bm{x})\leq\mathfrak{s}(\bm{x})\leq\bar{v}_{0}\cdot r_{2,\alpha}(\bm{x}),

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$ .

Fix $p\in H$ , we may choose local special holomorphic coordinates $\{w_{j}\}_{j=1}^{n-1}$ in some neighborhood of $p$ in $D$ such that that

(4.193)

\omega_{D}=\omega_{\mathbb{C}^{n-1}}+O(|w|),

where

(4.194)

\omega_{\mathbb{C}^{n-1}}\equiv\frac{\sqrt{-1}}{2}\sum_{j=1}^{n-1}dw_{j}\wedge d\bar{w}_{j}.

Then by the analysis in Section 4.1, one can see that

(4.195)

T^{\frac{2n-2}{n}}\omega_{T}=\Big(T\omega_{TN,T}\oplus T^{2}\omega_{\mathbb{C}^{n-2}}\Big)+T^{2}\pi^{*}(\omega_{D}-\omega_{\mathbb{C}^{n-1}})+TO^{\prime}(s^{3})+Td(s^{2}\Gamma)

where $\omega_{TN,T}$ is the Taub-NUT metric on $\mathbb{C}^{2}_{u_{1},u_{2}}$ given by (2.52), and

(4.196)

\omega_{\mathbb{C}^{n-2}}\equiv\frac{\sqrt{-1}}{2}\sum_{j=2}^{n-1}dw_{j}\wedge d\bar{w}_{j}.

Notice that, we have already used the relations

(4.197)

\pi^{*}(O^{\prime}(r^{p}))=O^{\prime}(s^{2p})(p\geq 1),\quad\pi^{*}dy=\widetilde{O}(s),\quad\pi^{*}dz=\widetilde{O}(s).

We perform a change of coordinates

(4.198)

z=T^{-1}\underline{z},\ y=T^{-1}\underline{y},\ w_{j}=T^{-1}\underline{w}_{j},\ u_{k}=T^{-1/2}\underline{u}_{k}

and denote

(4.199)

{\bm{w}}\equiv(\underline{w}_{2},\cdots,\underline{w}_{n-1}),\ {\bm{u}}\equiv(\underline{u}_{1},\underline{u}_{2}),\ \underline{s}=|{\bm{u}}|.

From now on, we write the tensors $\omega_{TN,1}$ and $\Omega_{TN,1}$ with respect to those rescaled coordinates $\bm{w}$ and $\bm{u}$ , we have

(4.200)

T\omega_{TN,T}\equiv\omega_{TN,1},\quad T^{2}\omega_{\mathbb{C}^{n-2}}\equiv\omega_{\mathbb{C}^{n-2}},

where “ $\equiv$ ” means that the two metrics are isometric. Moreover,

(4.201)

\begin{cases}T^{2}\pi^{*}(\omega_{D}-\omega_{\mathbb{C}^{n-1}})=O((|{\bm{w}}|+|{\bm{u}}|^{2})T^{-1})\\ TO^{\prime}(s^{3})=O(T^{-3/2}\underline{s}^{3})\\ Td(s^{2}\Gamma)=O(T^{-3/2}\underline{s}).\end{cases}

The above computations impies

(4.202)

|T^{\frac{2n-2}{n}}\omega_{T}-(\omega_{TN,1}\oplus\omega_{\mathbb{C}^{n-2}})|_{C^{2,\alpha}}=O(T^{-1}),

where the norm is measured with respect to the limiting product metric $\omega_{TN,1}\oplus\omega_{\mathbb{C}^{n-2}}$ .

In a similar vein, by the analysis in Section 4.1, we also obtain the expansion for the holomorphic form $\Omega_{T}$ ,

(4.203)

T^{n-1}\Omega_{T}=\Omega_{TN,1}\wedge d\underline{w}_{2}\wedge\cdots\wedge d\underline{w}_{n-1}+O((|{\bm{w}}|+|{\bm{u}}|^{2})T^{-1}),

which gives the convergence of $T^{n-1}\Omega_{T}$ .

Notice that, the above convergence is smooth away from $\bm{0}^{2}\times\mathbb{C}^{n-2}$ , where $\bm{0}^{2}\in\mathbb{C}_{TN}^{2}$ .

Starting from the above deepest bubble, we will let the reference point $\bm{x}$ keep away from the singular set $\mathcal{P}$ and switch to the next region where we will see that the bubbles transform from the Taub-NUT geometry to the cylindrical geometry. By definition, the reference point $\bm{x}$ in this region satisfies the relation

(4.204)

\frac{1}{2T}\leq r(\bm{x})\leq 1.

Region $\bf{I}_{2}$ (bubble transformations):

Figure 4.2. Bubble limits

\mathbb{C}_{TN,\varrho}^{2}\times\mathbb{C}^{n-2}

and

\mathbb{R}^{3}\times\mathbb{C}^{n-2}

in Region

\bf{I}_{2}

: The red circle is the

S^{1}

-fiber at the infinity of

\mathbb{C}_{TN,\varrho}^{2}

whose length equals

\frac{1}{\varrho}\geq(\sigma_{0})^{2}>0

;

\Sigma_{0^{3}}=\{0^{3}\}\times\mathbb{C}^{n-2}

is the singular set in

\mathbb{R}^{3}\times\mathbb{C}^{n-2}

and

d(\bm{x}_{\infty},\Sigma_{0^{3}})=1

Figure 4.3. Bubble limit

D\times\mathbb{R}

in Region

\bf{I}_{2}

. Here

P=H\times\{0\}

with

H\subset D

is the singular set in

D\times\mathbb{R}

In this region, the Kähler metric $\omega_{T}$ on $\mathcal{M}_{T}$ can be viewed as the lifting metric of the Riemannian submersion $\mathcal{M}_{T}\to Q\setminus P$ , i.e.,

(4.205)

g_{T}=T^{\frac{2-n}{n}}\cdot\Big(\pi^{*}(Tg_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}\Big),

where $g_{T}$ , $g_{0}$ and $g_{1}$ are the Riemannian metrics corresponding to the Kähler forms $\omega_{T}$ , $\omega_{D}$ and $\psi$ respectively.

As $r(\bm{x})$ varies from $2T^{-1}$ to $1$ , the Gromov-Hausdorff limit of the rescaled space $\Big(\mathcal{M}_{T},\mathfrak{s}(\bm{x})^{-2}g_{T},\bm{x}\Big)$ will correspondingly change (see Figure 4.2 and Figure 4.3). We will show that, for each $\bm{x}\in\bf{I}_{2}$ , the regularity scale is given by

(4.206)

\mathfrak{s}(\bm{x})=T^{\frac{1}{n}}\cdot r(\bm{x}).

More specifically, we will prove that under the rescaled metrics $\tilde{g}_{T}=(\mathfrak{s}(\bm{x}))^{-2}g_{T}$ , the Gromov-Hausdorff convergence keeps $\frac{1}{C_{0}}\leq|\Rm_{\tilde{g}_{T}}(\bm{x})|\leq C_{0}$ as $T\to+\infty$ ,

(4.207)

(\mathcal{M}_{T},\tilde{g}_{T},\bm{x})\xrightarrow{GH}(\mathcal{M}_{\infty},\tilde{d}_{\infty},\bm{x}_{\infty}).

Let $r_{j}\equiv r(\bm{x}_{j})$ , then we divide the region $\bf{I}_{2}$ into three disjoint pieces depending on the scale of $r_{j}$ , which will give different bubble limits (see Figure 4.2 and and Figure 4.3):

(a)

There is some $\sigma_{0}>0$ such that

(4.208) $c_{0}\cdot T_{j}^{-1}\leq r_{j}\leq\frac{1}{(\sigma_{0})^{2}}\cdot T_{j}^{-1}.$
(b)

Assume that $r_{j}$ satisfies the following condition holds,

(4.209) $\frac{r_{j}}{T_{j}^{-1}}\to\infty,\ r_{j}\to 0.$
(c)

Assume that there is some $\underline{T}_{0}>0$ such that

(4.210) $\underline{T}_{0}\leq r_{j}\leq 1.$

Case (a) is the same as Region $\bf{I}_{1}$ such that we have the $C^{2,\alpha}$ convergence of the spaces $(\mathcal{M}_{T},\tilde{g}_{j},\bm{x}_{j})$ towards the product space $\mathbb{C}^{2}_{TN,\varrho}\times\mathbb{C}^{n-2}$ , where

(4.211)

\varrho\equiv\lim_{j\rightarrow\infty}T_{j}\cdot r_{j}\in[c_{0},\frac{1}{\sigma_{0}^{2}}].

Therefore, if we choose $\mathfrak{s}(\bm{x})=T^{\frac{1}{n}}\cdot r(\bm{x})$ ,

(4.212)

\underline{v}_{0}\cdot r_{2,\alpha}(\bm{x})\leq\mathfrak{s}(\bm{x})\leq\bar{v}_{0}\cdot r_{2,\alpha}(\bm{x}),

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$ .

In the following calculations, we will rescale the coordinates as follows

(4.213)

z=r_{j}\cdot\underline{z},\ y=r_{j}\cdot\underline{y},\ w_{p}=r_{j}\cdot\underline{w}_{p},\ u_{q}=r_{j}^{1/2}\cdot\underline{u}_{q},

where $r_{j}\equiv r(\bm{x}_{j})$ , $p\in\{2,\ldots,n-1\}$ , $q\in\{1,2\}$ . For simplicity, we denote

(4.214)

s_{j}\equiv\mathfrak{s}(\bm{x}_{j})\quad\text{and}\quad\lambda_{j}\equiv s_{j}^{-1}.

Notice that, in Case (b) and Case (c), as $T\to+\infty$ , curvatures tend to infinity along the singular set $\mathcal{P}\subset\mathcal{M}_{T}$ , in the mean while, the rescaled distance $d_{\tilde{g}_{T}}(\bm{x}_{j},\mathcal{P})$ is uniformly bounded. Therefore, in the following, we will analyze both the convergence of the entire neck region $\mathcal{M}_{T}$ and the limiting behavior of the geometry bounded region which is a punctured region in $\mathcal{M}_{T}$ obtained by removing some small tubular neighborhood of $\mathcal{P}$ in $\mathcal{M}_{T}$ . For any $b>a>0$ , we denote

(4.215)

\mathfrak{U}_{j}(a,b)=\{\bm{x}\in\mathcal{M}_{T_{j}}|a\leq z(\bm{x})\leq b\}.

We will study the convergence of the punctured region

(4.216)

\mathring{\mathfrak{U}}_{j}\equiv\mathfrak{U}_{j}(z_{j}-\xi_{j},z_{j}+\xi_{j})\setminus\mathcal{S}_{j},

as $T_{j}\to+\infty$ , where $\mathcal{S}_{j}$ is a small neighborhood of $\mathcal{P}$ to be determined later.

Case (b):

First, we study Case (b) which is in fact the limiting case of Case (a) as $\sigma_{0}\to 0$ . Geometrically, the rescaled limit $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ in Case (b) is the asymptotic cone of the product space $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ which is isometric to the product Euclidean space $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ .

For an embedded submanifold $N\subset M$ , let us denote by $T_{r}(N)$ the $r$ -tubular neighborhood of $N$ in $M$ :

(4.217)

T_{r}(N)\equiv\{x\in Q|d_{M}(x,N)\leq r\}.

In this case, we choose the tubular neighborhood of $\mathcal{P}=\pi^{-1}(P)\subset\mathcal{M}_{T}$ ,

(4.218)

\mathcal{S}_{j}\equiv T_{r_{j}}(\mathcal{P},g_{j})

with respect to the original metrics $g_{j}\equiv g_{T_{j}}$ . Let $\xi_{j}$ satisfy $\xi_{j}\cdot T^{-\frac{1}{n}}\geq 1$ , then we will show that

(4.219)

(\mathring{\mathfrak{U}}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big((\mathbb{R}^{3}\times\mathbb{C}^{n-2})\setminus\Sigma_{0^{3}},g_{\mathbb{R}^{2n+1}},\bm{x}_{\infty}\Big).

where $\Sigma_{0^{3}}\equiv(\{0^{3}\}\times\mathbb{C}^{n-2})\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}=\mathbb{R}^{2n-1}$ and $d_{\mathbb{R}^{2n-1}}(\bm{x}_{\infty},\Sigma_{0^{3}})=1$ .

To start with, it is straightforward that under the rescaled metric $\tilde{g}_{j}$ ,

(4.220)

\widetilde{\mathcal{S}}_{j}=T_{\lambda_{j}r_{j}}(\mathcal{P},\tilde{g}_{j})

converges to a slice $\Sigma_{0^{3}}\equiv(\{0^{3}\}\times\mathbb{C}^{n-2})$ because $\lambda_{j}r_{j}=T_{j}^{-\frac{1}{n}}\to 0$ . Next, the limiting behavior of the rescaled metrics $\tilde{g}_{j}$ can be computed explicitly. Now we calculate the limit of each term in $\tilde{g}_{j}=T_{j}^{\frac{2-n}{n}}\cdot(\pi^{*}(T_{j}g_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}),$ which is given by (4.205): First, the scale assumption in Case (b) $r_{j}\to 0$ and $r_{j}T_{j}\to+\infty$ imply that

	$\displaystyle\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot\pi^{*}(T_{j}g_{0}+g_{1})$	$\displaystyle=r_{j}^{-2}\cdot T_{j}^{-1}\cdot\pi^{*}(T_{j}g_{0}+g_{1})$
		$\displaystyle=r_{j}^{-2}\pi^{}(g_{0})+r_{j}^{-2}\cdot T_{j}^{-1}\cdot\pi^{}(g_{1})$
(4.221)			$\displaystyle\to g_{\mathbb{C}^{n-1}},$

where we used the rescaled coordinates (4.213) in the computations. By the same computation,

(4.222)		$\displaystyle\lambda_{j}^{2}\cdot\pi^{*}\Big(T_{j}^{\frac{2-n}{n}}\cdot(h\cdot dz^{2})\Big)=(\Tr_{\omega_{D}}\psi+q(z))\cdot T_{j}^{-1}(d\underline{z})^{2}$	$\displaystyle\to g_{\mathbb{R}},$
(4.223)		$\displaystyle\lambda_{j}^{2}\cdot T^{\frac{2-n}{n}}\cdot(h^{-1}\Theta^{2})=r_{j}^{-2}\cdot T_{j}^{-1}\cdot(h^{-1}\Theta^{2})$	$\displaystyle\to 0.$

Therefore, we obtained the desired convergence.

Now that we have proved the convergence (4.219), so we will locally lift $B_{\frac{1}{2}}(\bm{x}_{j})$ to the universal cover $(\widetilde{B_{\frac{1}{2}}(\bm{x}_{j})},\tilde{\bm{x}}_{j})$ . By explicit computations, it has uniformly bounded $C^{k,\alpha}$ -geometry for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ . In fact, this can be seen from the higher order convergence of $\psi$ and $h$ in the above expressions. Therefore, if we choose $\mathfrak{s}(\bm{x}_{j})=T^{\frac{1}{n}}\cdot r(\bm{x}_{j})$ , then for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ ,

(4.224)

\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j})\leq\mathfrak{s}(\bm{x}_{j})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j}),

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$ .

Case (c):

We will prove that, for appropriately chosen parameters $\xi_{j}$ and $\mu_{j}$ , the rescaled limit of the punctured annulus

(4.225)

\mathring{\mathfrak{U}}_{j}\equiv\mathfrak{U}_{j}(z_{j}-\xi_{j},z_{j}+\xi_{j})\setminus\mathcal{S}_{j}

with $\mathcal{S}_{j}\equiv\pi^{-1}(T_{\mu_{j}}(P))$ is a punctured cylinder $Q\setminus P$ . That is, let $\xi_{j}$ and $\mu_{j}$ be a sequence of numbers satisfying the condition

(4.226)		$\displaystyle\frac{T^{\frac{1}{n}}}{\xi_{j}}$	$\displaystyle\to 0,\quad\frac{\xi_{j}}{T_{j}}\to 0,$
(4.227)		$\displaystyle\frac{1}{T_{j}\mu_{j}}$	$\displaystyle\to 0,\quad\mu_{j}\to 0,$

then we will show that

(4.228)

(\mathring{\mathfrak{U}}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big(Q\setminus P,g_{Q},\bm{x}_{\infty}\Big),

where $g_{Q}$ is a product metric on $Q\equiv D\times\mathbb{R}$ and $d_{g_{Q}}(\bm{x}_{\infty},P)=1$ .

To see this, we need to estimate the size of $\mathring{\mathfrak{U}}_{j}$ and the puncture $\pi^{-1}(T_{\mu_{j}}(P))$ as $T_{j}\to+\infty$ . By definition, when the reference point $\bm{x}_{j}$ is in Case (c), the distance to the divisor $r_{j}\equiv r(\bm{x}_{j})$ satisfies

(4.229)

\underline{T}_{0}\leq r_{j}\leq 1,

which implies the metric rescaling factor $\lambda_{j}$ satisfies

(4.230)

T_{j}^{-\frac{1}{n}}\leq\lambda_{j}\leq T_{j}^{-\frac{1}{n}}\cdot\underline{T}_{0}^{-1}.

Let $\lambda_{0}>0$ be a positive constant such that passing to a subsequence, $T_{j}^{\frac{1}{n}}\cdot\lambda_{j}\to\lambda_{0}$ . In the following, we will show that the limit of the rescaled metric

(4.231)

\tilde{g}_{j}=\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot\Big(\pi^{*}(Tg_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}\Big)

is the Riemann product

(4.232)

g_{Q}=\lambda_{0}^{2}(dz^{2}+g_{0}).

In fact, by the choice of $\mu_{j}$ , we have for every $\bm{y}\in\mathring{\mathfrak{U}}_{j}$ , $r(\bm{y})\geq\mu_{j}$ . Hence there is a smooth function $\chi$ satisfying $|\chi|=O^{\prime}(r)$ and $|\chi|\leq C\cdot\xi_{j}\ll T_{j}$ such that

(4.233)

\Big|h(\bm{y})-\Big(\chi_{j}(\bm{y})+T_{j}\Big)\Big|\leq\frac{1}{2\mu_{j}},

which implies

(4.234)

\Big|\frac{h(\bm{y})}{T_{j}}-1\Big|\leq\Big|\frac{h(\bm{y})}{T_{j}}-\Big(\frac{\chi_{j}(\bm{y})}{T_{j}}+1\Big)\Big|+\frac{|\xi_{j}(\bm{y})|}{T_{j}}\leq\frac{1}{2\mu_{j}T_{j}}+\frac{C\cdot\xi_{j}}{T_{j}}\to 0.

Therefore,

(4.235)

\displaystyle\Big|\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot h(\bm{y})-\lambda_{0}^{2}\Big|=\Big|(\lambda_{j}\cdot T^{\frac{1}{n}})^{2}\cdot\frac{h(\bm{y})}{T_{j}}-\lambda_{0}^{2}\Big|\to|\lambda_{0}^{2}-\lambda_{0}^{2}|=0.

Similarly, one can show that

(4.236)

\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot(Tg_{0}+g_{1})\to\lambda_{0}^{2}\cdot g_{0}.

Moreover, the above computations imply that $\mathfrak{U}_{j}$ has two ends and

(4.237)

\diam_{\tilde{g}_{j}}(\mathfrak{U}_{j})\approx C\cdot\xi_{j}\cdot T^{-\frac{1}{n}}\to\infty

and

(4.238)

\diam_{\tilde{g}_{j}}\Big(\pi^{-1}(T_{\mu_{j}}(P)\Big)\approx C\cdot\mu_{j}\cdot T_{j}^{-\frac{1}{n}}\to 0.

Therefore, applying (4.237), (4.238) and (4.235), we have

(4.239)

(\mathring{\mathfrak{U}}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big(Q\setminus P,g_{Q},\bm{x}_{\infty}\Big),

where $g_{Q}=\lambda_{0}^{2}(g_{0}+dz^{2})$ is the product metric on the cylinder $Q\times\mathbb{R}$ . Similar to Case (b), by choosing $\mathfrak{s}(\bm{x}_{j})=T^{\frac{1}{n}}\cdot r(\bm{x}_{j})$ , then for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ ,

(4.240)

\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j})\leq\mathfrak{s}(\bm{x}_{j})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j}),

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$ .

Now we care about the large scale geometries on $\mathcal{M}_{T}$ and let the reference point $\bm{x}$ keep far away from the singular set $\mathcal{P}$ . More precisely, we will focus on the region consisting of the points $\bm{x}$ satisfying

(4.241)

r(\bm{x})\geq\frac{1}{2}.

Region $\bf{I}_{3}$ (large scale geometries):

We will show that the regularity scale at each point $\bm{x}$ in this region is given by

(4.242)

\mathfrak{s}(\bm{x})=(L_{T}(\bm{x}))^{\frac{1}{2}}\cdot T^{\frac{2-n}{2n}}.

Moreover, we will calculate the rescaled limit with respect to each reference point in this region. Let $z_{j}\equiv z(\bm{x}_{j})$ , then depending upon the distance from the $\bm{x}_{j}$ to the singular set $\mathcal{P}$ , there are three cases to analyze:

(a)

(Close to the singular set $\mathcal{P}$ ) Assume that there is some $\zeta_{0}>0$ such that

(4.243) $r(\bm{x})\geq\frac{1}{2},\ |z_{j}|\leq\zeta_{0}.$
(b)

(Far from the singular set $\mathcal{P}$ and the boundary of $\mathcal{M}_{T}$ ) Assume that $z_{j}$ satisfies

(4.244) $|\zeta_{j}|\to\infty,\ \frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to 0.$

(c)

(Close to the boundary) Assume that there is some $c_{0}>0$ such that

(4.245)

\displaystyle c_{0}\leq\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\leq\begin{cases}c_{-},&\text{if}\ z_{j}<0,\\ c_{+},&\text{if}\ z_{j}>0.\end{cases}

Case (a) is identical to Case (c) of Region $\bf{I}_{1}$ such that the rescaled limit space is a cylinder $(Q,g_{Q},\bm{x}_{\infty})$ and $d_{g_{Q}}(\bm{x}_{\infty},P)\leq C_{0}$ for $P=H\times\{0\}\subset Q$ . Moreover, the convergence keeps curvatures uniformly bounded away from the singular set $P$ .

Case (b):

Now we switch to calculate the limiting metric in Case (b). In this case, with respect to the reference point $\bm{x}_{j}$ , the metric rescaling factor $\lambda_{j}\equiv\lambda(\bm{x}_{j})$ is chosen as

(4.246)

\lambda_{j}=(L_{T_{j}}(\bm{x}_{j}))^{-\frac{1}{2}}\cdot T_{j}^{\frac{n-2}{2n}}.

Let $\mathfrak{U}_{j}\equiv\mathfrak{U}(z_{j}-\xi_{j},z_{j}+\xi_{j})$ be the annulus centered at the slice $z=z_{j}$ such that

(4.247)

C^{-1}\cdot T_{j}^{\frac{n-2}{n}}\leq|\xi_{j}|\leq C\cdot T_{j}^{\frac{n-2}{n}}.

where $C>0$ is independent of $T_{j}$ . We will show that,

(4.248)

(\mathfrak{U}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(Q,g_{Q},\bm{x}_{\infty}).

In the following computations, we will also make appropriate coordinate change along the $\mathbb{R}$ -direction, that is, with respect to the reference point $\bm{x}_{j}$ , we pick coordinate $w$ such that

(4.249)

z=z_{j}+\Big(\frac{T_{j}}{L_{T_{j}}(z_{j})}\Big)^{\frac{n-2}{2}}w.

In the above notations, the rescaled metric $\tilde{g}_{j}$ can be represented as

	$\displaystyle\tilde{g}_{j}$	$\displaystyle=\lambda_{j}^{2}\cdot T_{j}^{\frac{n}{2-n}}\cdot\Big(\pi^{*}(Tg_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}\Big)$
(4.250)			$\displaystyle=L_{T_{j}}(z_{j})^{-1}\cdot\Big(\pi^{*}((Tg_{0}+g_{1})+hdz^{2})+h^{-1}\Theta^{2}\Big).$

Now we are in a position to work on the concrete expression of the limiting metric. Without loss of generality, we only consider the case $z_{j}=z(\bm{x}_{j})<0$ . Applying Lemma 3.31,

(4.251)

\displaystyle Tg_{0}+g_{1}=(k_{-}\cdot z(\bm{y})+\beta_{-}+T_{j})g_{0}+O(e^{-\delta\cdot z(\bm{y})}),

which implies that

	$\displaystyle L_{T_{j}}(z_{j})^{-1}\cdot\pi^{*}(Tg_{0}+g_{1})$	$\displaystyle=L_{T_{j}}(z_{j})^{-1}\cdot\Big(k_{-}z_{j}+\beta_{-}+T_{j}+k_{-}(z(\bm{y})-z_{j})\Big)$
(4.252)			$\displaystyle=1+\frac{k_{-}(z(\bm{y})-z_{j})+\beta_{-}}{L_{T_{j}}(z_{j})}.$

By (4.247) and (4.244), we have

(4.253)

\Big|\frac{k_{-}(z(\bm{y})-z_{j})+\beta_{-}}{L_{T_{j}}(z_{j})}\Big|\leq\frac{k_{-}|\xi_{j}|}{L_{T_{j}}(z_{j})}+\frac{\beta_{-}}{L_{T_{j}}(z_{j})}\to 0.

The above calculations imply that, as $T_{j}\to+\infty$ ,

(4.254)

L_{T_{j}}(z_{j})^{-1}\cdot\pi^{*}(Tg_{0}+g_{1})\to g_{0}.

Next, we compute the second term in (4.250),

	$\displaystyle L_{T_{j}}(z_{j})^{-1}\cdot\pi^{*}(h(\bm{y})\cdot dz^{2})$	$\displaystyle=L_{T_{j}}(z_{j})^{-1}\cdot T_{j}^{2-n}\cdot L_{T_{j}}(z(\bm{y}))^{n-1}\cdot\Big(\frac{T_{j}}{L_{T_{j}}(z_{j})}\Big)^{n-2}\cdot dw^{2}$
(4.255)			$\displaystyle=\Big(1+k_{-}\Big(\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\Big)^{\frac{n}{2}}\cdot w\Big)^{n-1}dw^{2}.$

In this case, the reference point $\bm{x}_{j}$ with $z_{j}=z(\bm{x}_{j})$ satisfies

(4.256)

\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to 0,

and hence as $T_{j}\to+\infty$ ,

(4.257)

L_{T_{j}}(z_{j})^{-1}\cdot\pi^{*}(h(\bm{y})\cdot dz^{2})\to dw^{2}.

Similarly,

(4.258)

L_{T_{j}}(z_{j})^{-1}\cdot h^{-1}\cdot\Theta^{2}\to 0.

Combining (4.254), (4.257) and (4.258), the rescaled limit is the product space $Q=D\times\mathbb{R}$ with the above limiting product metric

(4.259)

g_{Q}=g_{0}+dw^{2}.

In the above convergence, no singularity appears at all. Therefore, lifting to the universal cover, we have the $C^{k,\alpha}$ -convergence for $h$ and $\psi$ for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ , and hence by choosing $\mathfrak{s}(\bm{x}_{j})=(L_{T}(\bm{x}_{j}))^{\frac{1}{2}}\cdot T^{\frac{2-n}{2n}}$ , we have

(4.260)

\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j})\leq\mathfrak{s}(\bm{x}_{j})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j})

for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ , where $\underline{v}_{0}>0$ and $\bar{v}_{0}>0$ are uniform constants independent of $T\gg 1$ .

Case (c):

In this case, the reference point $\bm{x}_{j}$ is close to the boundary of $\mathcal{M}_{T}$ . The estimate (4.260) can be established in the same way. We only calculate the rescaled limit in the case $z_{j}<0$ . We will show that the rescaled limit is the incomplete Calabi space of complex dimension $n$ ,

(4.261)

(\mathcal{M}_{T},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{\infty}).

First, by the condition (4.245), there is some constant $\mathfrak{c}_{0}\in[c_{0},c_{-}]$ such that

(4.262)

\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to\mathfrak{c}_{0}.

Now check each term of the rescaled metric $\tilde{g}_{j}$ :

(4.263)	$\displaystyle L_{T_{j}}(z_{j})^{-1}\cdot(T_{j}\cdot g_{0}+g_{1})\to$	$\displaystyle(1+k_{-}\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot w)g_{0},$
(4.264)	$\displaystyle L_{T_{j}}(z_{j})^{-1}\cdot h(\bm{y})\cdot dz^{2}\to$	$\displaystyle(1+k_{-}\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot w)^{n-1}dw^{2},$
(4.265)	$\displaystyle L_{T_{j}}(z_{j})^{-1}\cdot(h^{-1}\Theta_{j}^{2})\to$	$\displaystyle\mathfrak{c}_{0}^{-n}\cdot(1+k_{-}\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot w)^{1-n}\Theta_{\mathcal{C}^{n}}^{2}.$

Therefore, $\tilde{g}_{j}$ converges to the Calabi metric

(4.266)

g_{\mathcal{C}_{-}^{n}}=(1+k_{-}\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot w)g_{0}+(1+k_{-}\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot w)^{n-1}dw^{2}+\mathfrak{c}_{0}^{-n}\cdot(1+k_{-}\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot w)^{1-n}.

Here $\Theta_{j}$ denotes the $S^{1}$ -connection of $\mathcal{M}_{T}$ , $\Theta_{\mathcal{C}^{n}}$ is the $S^{1}$ -connection of the Calabi space $\mathcal{C}^{n}$ , and the convergence holds up to some gauge transformations. Therefore, $\tilde{g}_{j}$ converges to the Calabi model metric. Moreover, up to the local universal cover, the above convergence is $C^{k,\alpha}$ for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$

In summary, we are led to unify the expression of the regularity scale for each $\bm{x}\in\mathcal{M}_{T}$ . For convenience, we slightly smoothing the distance function to $P$ as follows. Consider the cylinder $Q=D\times\mathbb{R}$ and let $r:Q\to\mathbb{R}_{+}\cup\{0\}$ be the distance to $P=H\times\{0\}$ . Then we are able to obtain a smooth function $\mathfrak{r}(\bm{x}):Q\to\mathbb{R}_{+}$ by slightly interpolating the distance function $r(\bm{x})$ in the overlapping regions of $\bf{I}_{1}$ , $\bf{I}_{2}$ , $\bf{I}_{3}$ such that $\mathfrak{r}(\bm{x})>0$ satisfies

(4.267)

\displaystyle\mathfrak{r}(\bm{x})=\begin{cases}T^{-1},&r(\bm{x})\leq T^{-1},\\ r(\bm{x}),&2T^{-1}\leq r(\bm{x})\leq\frac{1}{4},\\ 1,&r(\bm{x})\geq\frac{1}{2}.\end{cases}

Proposition 4.18 (Regularity scale on $\mathcal{M}_{T}$ ).

There are uniform constants $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ such that for each $\bm{x}\in M_{T}$ , the $C^{k,\alpha}$ -regularity scale $r_{k,\alpha}(\bm{x})$ at $\bm{x}$ has an explicit bound

(4.268)

\underline{v}_{0}\leq\frac{r_{k,\alpha}(\bm{x})}{\mathfrak{s}(\bm{x})}\leq\bar{v}_{0}.

The scale function $\mathfrak{s}(\bm{x})$ is expressed as follows,

(4.269)

\displaystyle\mathfrak{s}(\bm{x})=(\frac{L_{T}(\bm{x})}{T})^{\frac{1}{2}}\cdot\mathfrak{r}(\bm{x})\cdot T^{\frac{1}{n}},\quad\bm{x}\in\mathcal{M}_{T},

where $L_{T}(\bm{x})$ is defined in (4.12). Moreover, $k=2$ in Region $\bf{I}_{1}$ . In all other cases, $k$ is any positive integer.

Remark 4.18.1.

Notice that, the quotient $\frac{L_{T}(\bm{x})}{T}=1+O(T^{-1})$ as along as $|z(\bm{x})|$ is bounded.

Remark 4.18.2.

In the above computations, the key point in the collapsed cases is to reduce the metric convergence to the convergence of the harmonic function $h$ and the current $\psi$ by passing to the local universal cover. This can be done when we rescale the metric such that the $C^{k,\alpha}$ -geometry is uniformly bounded. In fact, this is exactly the reason why we introduce the notion of $C^{k,\alpha}$ -regularity scale.

Remark 4.18.3.

In the $4$ -dimensional case, the regularity scales were studied in Section 7 of [HSVZ18]. Mainly, we used lemma 7.2 and lemma 7.7 to deal with the special case with a limit $\mathbb{T}^{2}\times\mathbb{R}$ . Currently in the general case, we share the same spirit but the calculations are more technically involved.

Proposition 4.18 has an immediately corollary regarding the uniform Harnack type inequality for the regularity scale, which will be used in Section 4.4 for the weighted Schauder estimate.

Corollary 4.18.1 (Harnack inequality for the regularity scale).

There are some uniform constants $\underline{v}_{0}>0$ and $\overline{v}_{0}>0$ independent of $T\gg 1$ such that for each $\bm{x}\in\mathcal{M}_{T}$ , we have

(4.270)

\underline{v}_{0}\leq\frac{\mathfrak{s}(\bm{y}_{1})}{\mathfrak{s}(\bm{y}_{2})}\leq\overline{v}_{0}

for all $\bm{y}_{1},\bm{y}_{2}\in B_{\frac{\mathfrak{s}(\bm{x})}{4}}(\bm{x})$ .

The proof easily follows from the triangle inequality.

4.4. Fundamental estimates in the weighted Hölder spaces

Based on the above detailed studies of the regularity scales, we are ready to define the weighted Hölder space on the neck. To start with, let us recall the notation,

(4.271)		$\displaystyle\mathcal{M}_{T}$	$\displaystyle\equiv\Big\{\bm{x}\in\mathcal{M}\Big\|T_{-}\leq z(\bm{x})\leq T_{+}\Big\},$
(4.272)		$\displaystyle\mathring{\mathcal{M}}_{T}$	$\displaystyle\equiv\Big\{\bm{x}\in\mathcal{M}\Big\|T_{-}\leq z(\bm{x})\leq T_{+},\ d_{\omega_{T}}\Big(\bm{x},\partial\mathcal{M}_{T}\Big)\geq 1\Big\}.$

Based on the subdivision in Section 4.3, now we are able to define the weight functions and the weighted Hölder spaces.

Definition 4.19 (Weight function).

Given fixed real parameters $n\geq 2$ , $T>10^{3}$ , $\delta>0$ , $\nu,\mu\in\mathbb{R}$ and $\alpha\in(0,1)$ . For each $k\in\mathbb{N}$ , the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ is defined as follows,

(4.273)

\displaystyle\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=e^{\delta\cdot U_{T}(\bm{x})}\cdot\mathfrak{s}(\bm{x})^{\nu+k+\alpha}\cdot T^{\mu},

where $\mathfrak{s}(\bm{x})$ is the regularity scale at $\bm{x}$ given by Proposition 4.18 and

(4.274)		$\displaystyle U_{T}(\bm{x})$	$\displaystyle\equiv T\Big(1-(\frac{L_{T}(\bm{x})}{T})^{\frac{n}{2}}\Big),$
(4.275)		$\displaystyle L_{T}(\bm{x})$	$\displaystyle\equiv L_{T}(z(\bm{x}))=T+L_{0}(z(\bm{x})),$

where the functions $L_{T}$ and $L_{0}$ are defined in (4.12).

To better understand the weight function (4.273), we give several remarks.

Remark 4.19.1.

The function $e^{\delta\cdot U_{T}(\bm{x})}$ is the dominating term at large scales on $\mathcal{M}_{T}$ which behaves like an exponential function. The term $U_{T}(\bm{x})$ is defined by (4.275) just for unifying the weighted analysis for different “large scales” on $\mathcal{M}_{T}$ , which will be seen in the proof of Proposition 6.10 in Section 6. For intuition, there are two cases in which $U_{T}$ has simple expressions:

(4.276)

\displaystyle\begin{cases}U_{T}(\bm{x})=-L_{0}(z),&n=2,\\ U_{T}(\bm{x})\approx-\frac{n}{2}\cdot L_{0}(z),&n>2,\ |z(\bm{x})|\ll T.\end{cases}

Remark 4.19.2.

In the region $r(\bm{x})\leq 1/4$ , we can relate the distance function $d_{\mathcal{P}}(\bm{x})\equiv d_{\omega_{T}}(\bm{x},\mathcal{P})$ on $\mathcal{M}_{T}$ with $r(\bm{x})=d_{Q}(\pi(\bm{x}),P)$ as follows,

(4.277)

\displaystyle\begin{cases}C^{-1}\cdot T^{\frac{2-n}{2n}}\cdot r(\bm{x})^{1/2}\leq d_{P}(\bm{x})\leq C\cdot T^{\frac{2-n}{2n}}\cdot r(\bm{x})^{1/2},&r(\bm{x})\leq T^{-1},\\ C^{-1}\cdot T^{\frac{1}{n}}\cdot r(\bm{x})\leq d_{P}(\bm{x})\leq C\cdot T^{\frac{1}{n}}\cdot r(\bm{x}),&2T^{-1}\leq r(\bm{x})\leq\frac{1}{4}.\end{cases}

The weight function we used in [HSVZ18] was defined with respect to the intrinsic distance function $d_{P}(\bm{x})$ . Noticing by (4.277), the weight function defined by (4.273) essentially coincides with the one in [HSVZ18] (see Section 8 in [HSVZ18]).

Remark 4.19.3.

The constant term $T^{\mu}$ in the definition of the weight function is needed to deal with the non-linear term in the application of the implicit function theorem (see Proposition 6.4). When $n=2$ the non-linear term is quadratic and this constant term is unnecessary, but when $n>2$ we need to choose appropriate $\mu$ so that the weight function has a uniform lower bound independent of $T$ .

Lemma 4.20 (Lower bound estimate for the weight function).

For fixed constants $\delta>0$ , $\mu,\nu\in\mathbb{R}$ , $\alpha\in(0,1)$ and $k\in\mathbb{N}$ , then for all $T\gg 1$ and $\bm{x}\in\mathcal{M}_{T}$ ,

(4.278)

\displaystyle\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq\begin{cases}T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu},&\nu+k+\alpha\geq 0,\\ T^{\frac{\nu+k+\alpha}{n}+\mu},&\nu+k+\alpha<0.\end{cases}

Proof.

This lower bound estimate can be obtained by analyzing the regularity scale $\mathfrak{s}(\bm{x})$ . Denote by $w\equiv\frac{L_{T}(\bm{x})}{T}$ and recall that the two end points $T_{-},T_{+}$ satisfy

(4.279)

\displaystyle\begin{cases}L_{T}(T_{-})=T^{\frac{n-2}{n}}\\ L_{T}(T_{+})=T^{\frac{n-2}{n}},\end{cases}

then we have $w\in[T^{-\frac{2}{n}},1]$ . So it follows that

(4.280)

\rho_{\delta,\nu,\mu}^{(k+\alpha)}=F(w)\cdot\mathfrak{r}(\bm{x})^{\nu+k+\alpha}\cdot T^{\frac{\nu+k+\alpha}{n}+\mu},

where $F(w)\equiv e^{\delta\cdot T(1-w^{\frac{n}{2}})}\cdot w^{\frac{\nu+k+\alpha}{2}}$ . By the definition of $\mathfrak{r}(\bm{x})$ , immediately we have

(4.281)

\displaystyle T^{-1}\leq\mathfrak{r}(\bm{x})\leq 1

for all $\bm{x}\in\mathcal{M}_{T}$ , so it follows that

(4.282)

\displaystyle\rho_{\delta,\nu,\mu}^{(k+\alpha)}\geq\begin{cases}F(w)\cdot T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu},&\nu+k+\alpha\geq 0,\\ F(w)\cdot T^{\frac{\nu+k+\alpha}{n}+\mu},&\mu+\nu+k+\alpha<0,\end{cases}

Now it suffices to compute the lower bound of $F(w)$ . To this end, there are two cases to analyze depending on the sign of $\nu+k+\alpha$ . First, let $\nu+k+\alpha\leq 0$ , then obviously $F(w)\geq F(1)=1$ and hence

(4.283)

\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu}.

Next, we consider the case $\nu+k+\alpha>0$ . Simple calculus shows that $F(w)$ achieves its minimum in $[T^{-\frac{2}{n}},1]$ either at $w=1$ or at $w=T^{-\frac{2}{n}}$ . Notice that $F(T^{-\frac{2}{n}})\gg F(1)$ as $T\gg 1$ . This tells us that

(4.284)

\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq T^{\frac{\nu+k+\alpha}{n}+\mu}.

The proof is done.

∎

Using the above weight function, we define weighted Hölder spaces as follows.

Definition 4.21 (Weighted Hölder space).

Let $\mathcal{K}\subset\mathcal{M}_{T}$ be compact, then the weighted Hölder norm of a tensor field $\chi\in T^{r,s}(\mathcal{K})$ of type $(r,s)$ is defined by,

(4.285)		$\displaystyle\\|\chi\\|_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})}$	$\displaystyle\equiv\sum\limits_{m=0}^{k}\Big\\|\rho_{\delta,\nu,\mu}^{(m)}\cdot\nabla^{m}\chi\Big\\|_{C^{0}(\mathcal{K})}+[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})},$
(4.286)		$\displaystyle[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})}$	$\displaystyle\equiv\sup_{\begin{subarray}{c}d_{g}(x,y)\leq\iota_{0}\\ x,y\in\mathcal{K}\end{subarray}}\Big\{\min\{\rho_{\delta,\nu,\mu}^{(k+\alpha)}(x),\rho_{\delta,\nu,\mu}^{(k+\alpha)}(y)\}\cdot\frac{\|\nabla^{k}\chi(x)-\nabla^{k}\chi(y)\|}{(d_{g}(x,y))^{\alpha}}\Big\},$

where $\iota_{0}\equiv\frac{1}{4}\InjRad_{g}(\mathcal{M})$ . In the above definition, the difference of the two covariant derivatives is defined in terms of the parallel translation along the minimal geodesic.

Remark 4.21.1.

By definition, it is direct to see

(4.287)

\|\chi\|_{C_{\delta,\nu,\mu}^{k}(\mathcal{K})}=\sum\limits_{m=0}^{k}\|\nabla^{m}\chi\|_{C_{\delta,\nu+m,\mu}^{0}(\mathcal{K})}.

With the above definition of the weighted Hölder space, we are ready to give a local uniform weighted Schauder estimate with respect to the Laplacian on the neck $(\mathcal{M}_{T},\omega_{T})$ .

Proposition 4.22 (Weighted Schauder estimate, the local version).

For every sufficiently large parameter $T\gg 1$ , let $\mathcal{M}_{T}$ be the neck region with an $S^{1}$ -invariant Kähler metric $\omega_{T}$ constructed in Section 4.1. Then the following estimates hold:

(1)

(Interior estimate) Given $k\in\{0,1\}$ and $\alpha\in(0,1)$ , there is some uniform constant $C_{k,\alpha}>0$ such that for any $\bm{x}\in\mathring{\mathcal{M}}(T_{-},T_{+})$ , $r\in(0,1)$ , $u\in C^{k+2,\alpha}(B_{s(\bm{x})}(\bm{x}))$ ,

		$\displaystyle r^{k+2+\alpha}\cdot\\|u\\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{r\cdot s(\bm{x})}(\bm{x}))}$
(4.288)		$\displaystyle\leq$	$\displaystyle C_{k,\alpha}\Big(\\|\Delta u\\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{2r\cdot s(\bm{x})}(\bm{x}))}+\\|u\\|_{C_{\delta,\nu,\mu}^{0}(B_{2r\cdot s(\bm{x})}(\bm{x}))}\Big),$

where $s(\bm{x})\equiv\frac{\mathfrak{s}(\bm{x})}{4}$ and $\mathfrak{s}(\bm{x})$ is the regularity scale at $\bm{x}$ given by Proposition 4.18.

(2)

(Higher order estimate away from $\mathcal{P}$ ) There exists some large constant $C_{\mathcal{P}}>0$ such that if $\bm{x}\in\mathring{\mathcal{M}}_{T}$ satisfies

(4.289) $r(\bm{x})\geq C_{\mathcal{P}}\cdot T^{-1},$

then the uniform Schauder estimate (4.288) holds for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ .

(3)

(Boundary estimate) For any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ , there exists some uniform constant $C_{k,\alpha}>0$ such that for all $\bm{x}\in\partial\mathcal{M}_{T}$ , $r\in(0,1)$ and $u\in C^{k+2,\alpha}(T_{2}(\partial\mathcal{M}_{T}))$ ,

		$\displaystyle r^{k+2+\alpha}\cdot\\|u\\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{r\cdot s(\bm{x})}^{+}(\bm{x}))}$
(4.290)		$\displaystyle\leq$	$\displaystyle C_{k,\alpha}\Big(\\|\Delta u\\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{2r\cdot s(\bm{x})}^{+}(\bm{x}))}+\Big\\|\frac{\partial u}{\partial n}\Big\\|_{C_{\delta,\nu,\mu}^{k+1,\alpha}(B_{2r\cdot s(\bm{x})}^{+}(\bm{x}))}+\\|u\\|_{C_{\delta,\nu,\mu}^{0}(B_{2r\cdot s(\bm{x})}^{+}(\bm{x}))}\Big),$

where $B_{s}^{+}(\bm{x})\equiv B_{s}(\bm{x})\cap\mathcal{M}_{T}$ .

Remark 4.22.1.

The estimates (4.288) and (4.290) are not scale invariant. Notice that the scale parameter $r$ is always uniformly bounded from below in our actual applications. So both (4.288) and (4.290) are sufficient for our purpose.

Proof.

The main part is to prove Item (1). We only prove the estimate by assuming the scale parameter $r=1$ . The estimate in the general case $r\in(0,1)$ can be achieved by simple rescaling.

The proof is based on the explicit description of the $C^{k,\alpha}$ -regularity scale given by Proposition 4.18. Since we have shown that, under the rescalings

(4.291)

\displaystyle\tilde{g}=\lambda(\bm{x})^{2}\cdot g,

the geodesic balls $B_{1/2}^{\tilde{g}}(\bm{x})$ have uniformly bounded $C^{k,\alpha}$ -geometry (independent of $T$ ) for each $\alpha\in(0,1)$ and $k\in\{0,1\}$ . So there is a uniform constant $C>0$ (independent of $T$ ) such that the standard Schauder estimate holds for every $u\in\mathfrak{A}$ and $\bm{x}\in\mathring{\mathcal{M}}(T_{-},T_{+})$ ,

(4.292)

\|u\|_{C^{k+2,\alpha}(B_{1/4}^{\tilde{g}}(\bm{x}))}\leq C\Big(\|\Delta_{\tilde{g}_{j}}u\|_{C^{k,\alpha}(B_{1/2}^{\tilde{g}}(\bm{x}))}+\|u\|_{C^{0}(B_{1/2}^{\tilde{g}}(\bm{x}))}\Big).

Then the desired weighted Schauder estimate (4.288) will be obtained after appropriately rescaling. The argument is rather standard. In fact, the only crucial point is to verify that for every $\bm{x}\in\mathring{\mathcal{M}}(T_{-},T_{+})$ , the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ is roughly a constant in the ball $B_{s(\bm{x})}(\bm{x})$ in the sense that there is a uniform constant $C>0$ such that for any $\bm{y}\in B_{s(\bm{x})}(\bm{x})$ ,

(4.293)

C^{-1}\cdot\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\leq\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y})\leq C\cdot\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x}).

The verifications of the above estimate essentially follows from Corollary 4.18.1 which is the Harnack inequality for the regularity scale. As a comparison, the detailed arguments in dimension $4$ is given in Section 8 of [HSVZ18]. In the following, we only verify (4.293) in Region $\bf{I}_{1}$ and Region $\bf{I}_{2}$ as sample examples.

Region $\bf{I}_{1}$ :

By Proposition 4.18, the canonical scale in this case is chosen as $\mathfrak{s}(\bm{x})=T^{\frac{1-n}{n}}$ , while the rescaling factor is $\lambda(\bm{x})=T^{\frac{n-1}{n}}$ such that $(\mathcal{M}_{T},\tilde{g}_{T},\bm{x})$ is close to the Riemann product $\mathbb{C}_{TN,1}^{2}\times\mathbb{C}^{n-2}$ in the pointed $C^{2,\alpha}$ -topology for any $\alpha\in(0,1)$ , where $\mathbb{C}_{TN,1}^{2}$ is the Ricci-flat Taub-NUT space. Then for $k\in\{0,1\}$ and $\alpha\in(0,1)$ ,

(4.294)

\|u\|_{C^{k,\alpha}(B_{1}^{\tilde{g}_{T}}(\bm{x}))}\leq\|\Delta u\|_{C^{0,\alpha}(B_{2}^{\tilde{g}_{T}}(\bm{x}))}+\|u\|_{C^{0}(B_{2}^{\tilde{g}_{T}}(\bm{x}))}.

Since the weight function, by definition, is constant in the geodesic ball $B_{s(\bm{x})}(\bm{x})$ for $s(\bm{x})=\frac{1}{4}\mathfrak{s}(\bm{x})$ . With respect to the original metric, the standard Schauder estimate (4.292) for $u$ is equivalent to

(4.295)

\displaystyle\begin{split}&\sum\limits_{m=0}^{k+2}\|\rho_{\delta,\nu,\mu}^{(m)}\cdot\nabla^{m}u\|_{C^{0}(B_{s(\bm{x})}(\bm{x}))}+[\rho_{\delta,\nu,\mu}^{(k+2+\alpha)}\cdot\nabla^{k+2}u]_{C^{\alpha}(B_{s(\bm{x})}(\bm{x}))}\\ \leq&C\Big(\|\rho_{\delta,\nu+2,\mu}^{(0)}\cdot\Delta u\|_{C^{0}(B_{2s(\bm{x})}(\bm{x}))}+[\Delta u]_{C_{\delta,\nu+2,\mu}^{0,\alpha}(B_{2s(\bm{x})}(\bm{x}))}+\|\rho_{\delta,\nu,\mu}^{(0)}\cdot u\|_{C^{0}(B_{2s(\bm{x})}(\bm{x}))}\Big).\end{split}

Therefore, by the definition of the weighted Hölder space,

(4.296)

\displaystyle\|u\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{s(\bm{x})}(\bm{x}))}\leq C\Big(\|\Delta u\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(B_{2s(\bm{x})}(\bm{x}))}+\|u\|_{C_{\delta,\nu,\mu}^{0}(B_{2s(\bm{x})}(\bm{x}))}\Big).

The proof in Region $\bf{I}_{1}$ is done.

Region $\bf{I}_{2}$ :

Proposition 4.18 tells us that, in this region, $\mathfrak{s}(\bm{x})=T^{\frac{1}{n}}\cdot r(\bm{x})$ and the metric is rescaled by $\lambda(\bm{x})=\mathfrak{s}(\bm{x})^{-1}$ with

(4.297)

\tilde{g}=\lambda(\bm{x})^{2}g.

We notice that the values $\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y})$ for all $\bm{y}\in B_{2s(\bm{x})}(\bm{x})$ are uniformly equivalent. Indeed, by Corollary 4.18.1, we can see that for every $\bm{y}\in B_{2s(\bm{x})}(\bm{x})$ ,

(4.298)

\frac{1}{2}(\underline{v}_{0})^{\nu+k+\alpha}\leq\frac{\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y})}{\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})}\leq\frac{3}{2}(\overline{v}_{0})^{\nu+k+\alpha}.

So the standard Schauder estimate (4.292) for $\tilde{u}$ is equivalent to the following estimate for $u$ , is equivalent to

		$\displaystyle\sum\limits_{m=0}^{k+2}\\|\rho_{\delta,\nu,\mu}^{(m)}\cdot\nabla^{m}u\\|_{C^{0}(B_{s(\bm{x})}(\bm{x}))}+\\|\rho_{\delta,\nu,\mu}^{(k+2+\alpha)}\cdot\nabla^{k+2}u\\|_{C^{0,\alpha}((B_{s(\bm{x})}(\bm{x})))}$
(4.299)		$\displaystyle\leq$	$\displaystyle C\Big(\\|\rho_{\delta,\nu+2,\mu}^{(0)}\cdot\Delta u\\|_{C^{0}(B_{2s(\bm{x})}(\bm{x}))}+[\Delta u]_{C_{\delta,\nu+2,\mu}^{0,\alpha}(B_{2s(\bm{x})}(\bm{x}))}+\\|\rho_{\delta,\nu,\mu}^{(0)}\cdot u\\|_{C^{0}((B_{2s(\bm{x})}(\bm{x})))}\Big).$

Therefore, by the definition of the weighted norm, the required estimate immediately follows.

For the remaining regions, the key point in the proof is in fact the same, which just requires to show that the values of the weight function at the points within the $C^{k,\alpha}$ -regularity scale are uniformly equivalent. So we just skip the proof.

Now we switch to prove Item (2), which can be obtained by contradiction. Suppose there is no such a constant $C_{\mathcal{P}}>0$ . Then there are a sequence of numbers $T_{j}>0$ and reference points $\bm{x}_{j}\in\mathcal{M}_{T_{j}}$ such that

(4.300)

r(\bm{x}_{j})\cdot T_{j}\to+\infty,

but the uniform local Schauder estimate (4.288) does not hold around $\bm{x}_{j}\in\mathcal{M}_{T_{j}}$ . Under the contradicting assumption (4.300), Proposition 4.18 shows that, with respect to the rescaled metrics we choose, we will obtain one of the following rescaled Gromov-Hausdorff limits depending upon the location of $\bm{x}_{j}$ in the subdivision:

(i)

The Euclidean product $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ ,
(ii)

The cylinder $D\times\mathbb{R}$ ,
(iii)

The Calabi space $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{-})$ or $(\mathcal{C}_{+}^{n},g_{\mathcal{C}_{+}^{n}},\bm{x}_{+})$ .

Moreover, away from the singularity, the convergence is $C^{k,\alpha}$ for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ by passing to the local universal cover.

First, if the convergence keeps the $C^{k,\alpha}$ -geometry uniformly bounded, then the proof of the higher order estimate is just standard and routine.

Now let $\bm{x}_{j}$ stay in the regions giving the rescaled limits in (i) and (ii). Recall the discussions in Section 4.3 that, in Case (b), (c) in Region $\bf{I}_{2}$ and Case (a) in Region $\bf{I}_{3}$ , singularity behavior appears in the Gromov-Hausdorff procedure. With respect to the rescaled metric $\tilde{g}_{j}=\lambda(\bm{x}_{j})^{-2}g_{j}$ , the limiting geodesic ball $B_{\frac{1}{2}}^{\tilde{g}_{j}}(\bm{x}_{j})$ never contains the singularity. So it follows that every point $\bm{y}\in B_{\frac{1}{2}}^{\tilde{g}_{j}}(\bm{x}_{j})$ has a $C^{k,\alpha}$ -regularity scale $r_{k,\alpha}(\bm{y})\geq\rho_{0}>0$ for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ . So the standard interior Schauder estimate reads as follows,

(4.301)

\|u\|_{C^{k+2,\alpha}(B_{\frac{1}{4}}(\bm{x}_{j}))}\leq C_{k,\alpha}\cdot\Big(\|\Delta u\|_{C^{k,\alpha}(B_{\frac{1}{2}}(\bm{x}_{j}))}+\|u\|_{C^{k}(B_{\frac{1}{2}}(\bm{x}_{j}))}\Big),

for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ . Rescaling back to the original metrics $g_{j}$ , we obtain the desired weighted Schauder estimate for sufficiently large $j$ . So the contradiction arises. This completes the proof of Item (2).

The proof of Item (3) follows from the Schauder estimate for Neumann boundary problem. As before, we only consider the case $r=1$ for simplicity. The tubular neighborhood $T_{2}(\partial\mathcal{M}_{T})$ belongs to Case (c) of Region $\bf{I}_{3}$ . We only consider the left boundary $\{T=T_{-}\}$ . For every $\bm{x}\in\{T=T_{-}\}$ , we choose the rescaled metric $\tilde{g}=\lambda(\bm{x})^{2}\cdot g$ with

(4.302)

\lambda(\bm{x})=(L_{T}(T_{-}))^{-\frac{1}{2}}\cdot T^{\frac{n-2}{2n}}=(c_{-})^{\frac{1}{2}},

where $c_{-}>0$ is a fixed constant. The analysis in Section 4.3 tells us that, for $T\gg 1$ sufficiently large, $(\mathcal{M}_{T},\tilde{g},\bm{x})$ is Gromov-Hausdorff close to a fixed incomplete Calabi space $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{\infty})$ . Moreover, the tubular neighborhood $T_{2}(\partial M_{T})$ satisfies the following property: there are constants $\rho_{0}>0$ depending only the conjugate radius of $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{\infty})$ such that every point $\bm{y}\in T_{2}(\partial M_{T})$ satisfies the regularity scale estimate $r_{k,\alpha}(\bm{y})\geq\rho_{0}>0$ for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ .

The above geometric regularity implies the following uniform boundary Schauder estimate in $B_{2}^{+}(\bm{x})$ for each $\bm{x}\in\partial\mathcal{M}_{T}$ ,

(4.303)

\displaystyle\|u\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{\frac{1}{4}}^{+}(\bm{x}))}\leq C_{k,\alpha}\Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{\frac{1}{2}}^{+}(\bm{x}))}+\Big\|\frac{\partial u}{\partial n}\Big\|_{C_{\delta,\nu,\mu}^{k+1,\alpha}(B_{\frac{1}{2}}^{+}(\bm{x}))}+\|u\|_{C_{\delta,\nu,\mu}^{0}(B_{\frac{1}{2}}^{+}(\bm{x}))}\Big),

Here $\frac{\partial}{\partial n}$ is the exterior normal vector field, and the constant $C_{k,\alpha}>0$ depends only on $k$ , $\alpha$ , $\rho_{0}$ . This estimate is standard in the literature (see Section 6 of [GT01] for instance). By rescaling, we obtain the desired weighted estimate.

∎

We finish this subsection with the following weighted error estimate for the Calabi-Yau equation.

Proposition 4.23 (Weighted error estimate).

Let $\mathrm{Err}_{CY}$ be the error function given by Definition 4.3. For fixed parameters $\delta>0$ , $\mu,\nu\in\mathbb{R}$ and $\alpha\in(0,1)$ which satisfy

(4.304)		$\displaystyle 0<\delta<\delta_{e}$	$\displaystyle\equiv\frac{\sqrt{\lambda_{1}}}{n(\|k_{-}\|+\|k_{+}\|)},$
(4.305)		$\displaystyle\nu+\alpha$	$\displaystyle>0,$

where the constants $\lambda_{1}>0$ , $k_{-}>0$ and $k_{+}<0$ are given in Proposition 3.31. Then the weighted $C^{0,\alpha}$ -estimate holds,

(4.306)

\|\mathrm{Err}_{CY}\|_{C^{0,\alpha}_{\delta,\nu,\mu}(\mathcal{M}_{T})}=O(T^{-2+\frac{\nu+\alpha}{n}+\mu}).

Proof.

We again divide into different regions and estimate separately.

For $|z(\bm{x})|\leq 1$ , applying Corollary 3.24.1, we have

(4.307)

(\omega_{D}+T^{-1}\psi)^{n-1}=\omega_{D}^{n-1}(1+T^{-1}\Tr_{\omega_{D}}\psi+\sum_{k\geq 2}T^{-k}\Phi_{k})

where $\Phi_{k}=O^{\prime}(r^{k-1})$ is independent of $T$ . By (4.20) we have

(4.308)

T^{-1}h=1+T^{-1}\Tr_{\omega_{D}}\psi+T^{-2}\underline{B}(z).

Using (3.316), it is easy to see that

(4.309)

\|\mathrm{Err}_{CY}\|_{C^{0}(\{|z(\bm{x})|\leq 1\})}=O(T^{-2}),

Immediately, by the definition of the weighted $C^{0}$ -norm, we have

(4.310)

\|\mathrm{Err}_{CY}\|_{C_{\delta,\nu,\mu}^{0}(\{|z(\bm{x})|\leq 1\})}=O(T^{-2+\frac{\nu}{n}+\mu}),

Now consider the region $|z(\bm{x})|\geq 1$ , then by (3.349) we may write

(4.311)

\displaystyle\psi=\begin{cases}(k_{-}z)\cdot\omega_{D}+\xi,&z\leq-1,\\ (k_{+}z)\cdot\omega_{D}+\xi,&z\geq 1,\end{cases}

where $\xi=\epsilon(z)$ . So it follows that

	$\displaystyle(\omega_{D}+T^{-1}\psi)^{n-1}$	$\displaystyle=\Big((1+T^{-1}k_{\pm}z)\omega_{D}+T^{-1}\xi\Big)^{n-1}$
(4.312)			$\displaystyle=\omega_{D}^{n-1}\Big((1+T^{-1}k_{\pm}z)^{n-1}+(1+T^{-1}k_{\pm}z)^{n-2}T^{-1}\Tr_{\omega_{D}}\xi+O(T^{-2})\Big)$

By (4.16), we have

(4.313)

T^{-1}h=(1+T^{-1}k_{\pm}z)^{n-1}+T^{-1}\Tr_{\omega_{D}}\xi.

So we obtain

(4.314)

\mathrm{Err}_{CY}=\Big((1+T^{-1}k_{\pm}z)^{-1}-(1+T^{-1}k_{\pm}z)^{-n+1}\Big)T^{-1}\Tr_{\omega_{D}}\xi+O(T^{-2})

Since for $z\in[T_{-},T_{+}]$ ,

(4.315)

U_{T}(z)=T-T^{-\frac{n-2}{2}}(T+k_{\pm}z)^{\frac{n}{2}}=T(1-(1+T^{-1}k_{\pm}z)^{\frac{n}{2}})\leq-\frac{n}{2}\cdot k_{\pm}z.

Here we use the following elementary inequality: $(1-x)^{p}\geq 1-px$ for any $p\geq 1$ and $x\in(0,1)$ . By Proposition 3.31, the asymptotics $\xi=\epsilon(z)$ has the explicit exponential decaying rate $\epsilon(z)=O(e^{-(1-\tau)\sqrt{\lambda_{1}}z})$ for any $\tau\in(0,1)$ . Applying (4.315) and the the assumption

(4.316)

0<\delta<\delta_{e}\equiv\frac{\sqrt{\lambda_{1}}}{n(|k_{-}|+|k_{+}|)},

we conclude that, as $|z(\bm{x})|\to+\infty$ , the growth rate of $e^{\delta U_{T}(z(\bm{x}))}$ is slower than the decaying rate of $\epsilon(z)$ .

Therefore,

(4.317)

\|\mathrm{Err}_{CY}\|_{C^{0}(\{\|z(\bm{x})\|\geq 1\})}=O(T^{-2}).

By the definition of the weighted norm, we have

(4.318)

\|\mathrm{Err}_{CY}\|_{C_{\delta,\nu,\mu}^{0}(\{\|z(\bm{x})\|\geq 1\})}=O(T^{-2+\frac{\nu}{n}+\mu}).

The weighted $C^{0,\alpha}$ -estimate can be obtained in a similar way. It suffices to analyze the Hölder regularity around the singular set $\mathcal{P}$ . Notice that a fixed function in $O^{\prime}(r)$ has bounded $C^{0,\alpha}$ norm, so the weighted $C^{0,\alpha}$ -estimate is given by

(4.319)

\|\mathrm{Err}_{CY}\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M}_{T})}=O(T^{-2+\frac{\nu+\alpha}{n}+\mu}).

∎

4.5. Perturbation of complex structures

In Section 4.2 we have identified the underlying complex manifold of our family of $C^{2,\alpha}$ Kähler metrics $(\mathcal{M}_{T},\omega_{T})$ . In our gluing argument in Section 7.3 we shall need to perturb the complex structure. This section is devoted to the estimate of error caused by such a perturbation.

Under the holomorphic embedding of $\mathcal{M}_{T}$ into $\mathcal{N}^{0}$ defined in Section 4.2, $\Omega_{T}$ is identified with the standard holomorphic volume form $\Omega_{\mathcal{N}^{0}}$ .

Fix $C>0$ , and let $\mathcal{V}$ be the open neighborhood of $\mathcal{P}$ in $\mathcal{N}^{0}$ defined by $\{r_{+}<C,r_{-}<C\}$ . Fix a smooth Kähler metric $\omega_{\mathcal{N}^{0}}$ on $\mathcal{N}^{0}$ . Suppose now that we have a family of complex structures $J_{T}^{\prime}$ on $\mathcal{V}$ with holomorphic volume forms $\Omega_{T}^{\prime}$ satisfying for all $k\geq 0$ ,

(4.320)

\sup_{\bm{x}\in\mathcal{V}}|\nabla^{k}_{\omega_{\mathcal{N}^{0}}}(\Omega_{T}^{\prime}-\Omega_{\mathcal{N}^{0}})(\bm{x})|_{\omega_{\mathcal{N}^{0}}}\leq\underline{\epsilon}_{T^{2}}.

We also assume there is a deformation of the form $\pi^{*}\omega_{D}$ over $\mathcal{V}$ to $\omega_{D,T}$ , which is a closed $(1,1)$ form with respect $J_{T}^{\prime}$ , and satisfies that for all $k\geq 0$

(4.321)

\sup_{\bm{x}\in\mathcal{V}}|\nabla^{k}_{\omega_{\mathcal{N}^{0}}}(\omega_{D,T}-\pi^{*}\omega_{D})(\bm{x})|_{\omega_{\mathcal{N}^{0}}}\leq\underline{\epsilon}_{T^{2}}.

Let $\phi$ be the Kähler potential defined in (4.149). Then we define the new family of closed forms on $\mathcal{V}$

(4.322)

T^{\frac{n-2}{n}}\omega_{T}^{\prime}\equiv T\omega_{D,T}+dJ_{T}^{\prime}d\phi.

Proposition 4.24.

For $T$ sufficiently large, the above $(\omega_{T}^{\prime},\Omega_{T}^{\prime})$ defines a family of $C^{1,\alpha}$ Kähler structures on $\mathcal{V}$ , satisfying for all fixed $\alpha\in(0,1)$ , $\delta,\mu,\nu\in\mathbb{R}$ , we have

(4.323)		$\displaystyle\|\Omega_{T}^{\prime}-\Omega_{T}\|_{C^{2,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}$	$\displaystyle=\underline{\epsilon}_{T^{2}},$
(4.324)		$\displaystyle\|\omega_{T}^{\prime}-\omega_{T}\|_{C^{1,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}$	$\displaystyle=\underline{\epsilon}_{T^{2}}.$

We first reduce the estimate to a local form. Choose finitely many holomorphic charts $(U_{\beta},w_{1},\cdots,w_{n-1})$ in $D$ of the form $\{|w_{1}|<1,\cdots,|w_{n-1}|<1\}$ , such that the smaller charts given by $V_{\beta}=\{|w_{1}|<1/2,\cdots,|w_{n-1}|<1/2\}$ also cover $D$ . We may also assume if a $U_{\beta}$ intersects $H$ , then it is centered at some $p\in H$ , i.e. $w_{i}(p)=0$ for all $i$ , and also $H$ is defined by $w_{1}=0$ in this chart. We may further assume in each $U_{\beta}$ the line bundle $L$ has a holomorphic trivialization $\sigma_{\beta}$ , under which we may view $\zeta_{\pm}$ as local holomorphic functions on $\mathcal{N}^{0}$ , and $\mathcal{V}\cap\pi^{-1}(U_{\beta})$ is locally defined by $|\zeta_{\pm}|<C|\sigma_{L}|^{-|k_{\pm}|}$ . These then give an open cover of $\mathcal{V}$ by $\mathcal{V}\cap\pi^{-1}(V_{\beta})$ , and it suffices to prove the estimates in each such open set.

We shall work with one $\beta$ such that $U_{\beta}\cap H\neq\emptyset$ . The other case can be proved similarly. For such $\beta$ in $\pi^{-1}(U_{\beta})$ , by definition of $\mathcal{N}^{0}$ , we get the equation

(4.325)

\zeta_{+}\cdot\zeta_{-}=w_{1}F(w_{1},\cdots,w_{n-1})

for a non-zero holomorphic function $F$ . So without loss of generality we may assume $\zeta_{+},\zeta_{-},w_{2},\cdots,w_{n-1}$ are holomorphic coordinates on $\pi^{-1}(U_{\beta})$ .

We first prove (4.323). The hypothesis implies that

(4.326)

\Omega_{T}^{\prime}-\Omega_{T}=G_{i_{1}\ldots i_{n}}e_{i_{1}}\wedge\ldots\wedge e_{i_{n}},

where each $e_{j}$ is one of $d\zeta_{\pm},d\bar{\zeta}_{\pm},dw_{j},d\bar{w}_{j}(j\geq 2)$ , and $G_{i_{1}\ldots i_{n}}$ is a smooth function in $\zeta_{\pm},\bar{\zeta}_{\pm},w_{j},\bar{w}_{j}(j\geq 2)$ and its $k$ -th derivative over $\mathcal{V}$ with respect to the fixed metric $\omega_{\mathcal{N}^{0}}$ is bounded by $\underline{\epsilon}_{T^{2}}$ for all $k$ . Since a holomorphic function is automatically harmonic with respect to any Kähler metric, we have

(4.327)

\Delta_{\omega_{T}}\zeta_{\pm}=\Delta_{\omega_{T}}w_{j}=0.

By Corollary 4.11.1, Item (1), we know $\mathcal{V}$ is contained in the region $|z|\leq 1$ . Also notice by the discussion in Section 4.3 there is a constant $C>1$ such that for each $\bm{x}\in\mathcal{V}\cap\pi^{-1}(V_{\beta})$ , the ball $B_{C^{-1}\mathfrak{s}(\bm{x})}(\bm{x})$ is contained in $\pi^{-1}(U_{\beta})\cap\{|z|\leq 2\}$ . Again by Corollary 4.11.1, Item (3) on $\pi^{-1}(U_{\beta})\cap\{|z|\leq 2\}$ , we have

(4.328)

|\zeta_{\pm}|\leq Cr_{\pm}\leq Ce^{3T}.

Now applying Proposition 4.22 to every $\bm{x}\in\mathcal{V}\cap\pi^{-1}(V_{\beta})$ we obtain

(4.329)

|\zeta_{\pm}|_{C^{3,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}\leq C|\zeta_{\pm}|_{C^{0}_{\delta,\mu,\nu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}\leq Ce^{3T}.

Similarly, since on $U_{\beta}$ we have $|w_{j}|<1$ , we get for $j=2,\cdots,n-1$ ,

(4.330)

|w_{j}|_{C^{3,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}\leq|w_{j}|_{C^{0}_{\delta,\mu,\nu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}\leq C.

Then using the chain rule and induction we get that

(4.331)

|G_{i_{1}\cdots i_{n}}|_{C^{3,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}=\underline{\epsilon}_{T^{2}}\cdot Ce^{3T}=\underline{\epsilon}_{T^{2}}.

(4.332)

|\Omega_{T}^{\prime}-\Omega_{T}|_{C^{2,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}=\underline{\epsilon}_{T^{2}}.

Notice the complex structure $J_{T}^{\prime}$ is pointwise determined by the holomorphic $n$ form $\Omega_{T}^{\prime}$ algebraically, we get

(4.333)

|J_{T}^{\prime}-J_{T}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}=\underline{\epsilon}_{T^{2}}.

Now to prove (4.324), we write

(4.334)

\omega_{T}^{\prime}=\omega_{T}+T(\omega_{D,T}-\pi^{*}\omega_{D})+d((J_{T}^{\prime}-J_{T})d\phi).

By assumption, and the above discussion, using (4.329) we get

(4.335)

|\omega_{D,T}-\pi^{*}\omega|_{C^{2,\alpha}_{\delta,\nu,\mu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}=\underline{\epsilon}_{T^{2}}

It is also easy to see

(4.336)

|V_{1}\cdot V_{2}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq T^{m}|V_{1}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}|V_{2}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})},

for some $m>0$ independent of $V_{1}$ and $V_{2}$ . So (4.324) is a consequence of the following

Lemma 4.25.

(4.337)

|\phi|_{C^{3,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq e^{CT}.

Proof.

Since by construction

(4.338)

T^{\frac{n-2}{n}}\omega=T\pi^{*}\omega_{D}+dd^{c}\phi.

We have

(4.339)

\Delta_{T^{\frac{2n-2}{n}}\omega}\phi=n-T\cdot\Tr_{T^{\frac{2n-2}{n}}\omega}\pi^{*}\omega_{D}.

Since $\pi^{*}\omega_{D}$ is smooth on $\mathcal{V}$ and $\omega$ is parallel, again the above discussion gives that

(4.340)

|\Tr_{T^{\frac{2n-2}{n}}\omega}\pi^{*}\omega_{D}|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq e^{CT}.

So by Proposition 4.22 we get that

(4.341)

|\phi|_{C^{3,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq e^{CT}+C|\phi|_{C^{0}_{\delta,\nu,\mu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}.

To bound the right hand side we use the formula

(4.342)

\phi=\int_{T_{+}}^{z}uh(u)du+\phi(T_{+})=\int_{T_{+}}^{0}uh(u)du+\phi(T_{+})+\int_{0}^{z}uh(u)du.

Hence

(4.343)

\phi=\frac{T}{2}z^{2}+\frac{1}{2}r+B_{T}+O(1),

which gives

(4.344)

|\phi|_{C^{0}_{\delta,\nu,\mu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}\leq O(T^{m})

for some $m>0$ . The conclusion then follows. ∎

Remark 4.25.1.

In principle, it is possible to obtain more refined estimates with respect to the higher order weighted norms of $\zeta_{\pm}$ and $\phi$ by more direct calculation. The above argument using weighted Schauder estimates avoids the lengthy computations, and it suffices for our purpose since in our setting the error caused by complex structure perturbation is at the scale $e^{-CT^{2}}$ while the weighted analysis in the region $\{|z|\leq 1\}$ only introduces at most $e^{CT}$ error. It is also possible to improve the estimates by working on a scale much smaller than the regularity scale, but again that is not needed for our applications in this paper.

4. The approximately Calabi-Yau neck region [051E]

4. The approximately Calabi-Yau neck region

4.1. Construction of a family of C2,αC^{2,\alpha} Kähler structures

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Definition 4.3.

Remark 4.3.1.

Remark 4.3.2.

Lemma 4.4.

Proof.

Proposition 4.5.

Proof.

Remark 4.5.1.

Lemma 4.6.

Proof.

Proposition 4.7.

Proof.

4.2. Kähler geometry

4.2.1. The underlying complex manifold

Proposition 4.8.

Remark 4.8.1.

Proof.

Remark 4.8.2.

Lemma 4.9.

Proof.

Lemma 4.10.

Proof.

Proposition 4.11.

Proof.

Corollary 4.11.1.

Proof.

4.2.2. Kähler potentials

Proposition 4.12.

Remark 4.12.1.

Proof.

Remark 4.12.2.

Remark 4.12.3.

Lemma 4.13.

Proof.

4.3. Geometries at regularity scales

Definition 4.14 (Local regularity).

Definition 4.15 (Ck,αC^{k,\alpha}-regularity scale).

Example 4.16.

Example 4.17.

Remark 4.17.1.

Proposition 4.18 (Regularity scale on ℳT\mathcal{M}_{T}).

Remark 4.18.1.

Remark 4.18.2.

Remark 4.18.3.

Corollary 4.18.1 (Harnack inequality for the regularity scale).

4.4. Fundamental estimates in the weighted Hölder spaces

Definition 4.19 (Weight function).

Remark 4.19.1.

Remark 4.19.2.

Remark 4.19.3.

Lemma 4.20 (Lower bound estimate for the weight function).

Proof.

Definition 4.21 (Weighted Hölder space).

Remark 4.21.1.

Proposition 4.22 (Weighted Schauder estimate, the local version).

Remark 4.22.1.

Proof.

Proposition 4.23 (Weighted error estimate).

Proof.

4.5. Perturbation of complex structures

Proposition 4.24.

Lemma 4.25.

Proof.

Remark 4.25.1.

4.1. Construction of a family of $C^{2,\alpha}$ Kähler structures

Definition 4.15 ( $C^{k,\alpha}$ -regularity scale).

Proposition 4.18 (Regularity scale on $\mathcal{M}_{T}$ ).